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Abel.py
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Abel.py
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from functools import cache
from Binomial import binomial
from _tabltypes import MakeTriangle
"""Abel polynomials (unsigned coefficients).
[0] [1]
[1] [0, 1]
[2] [0, 2, 1]
[3] [0, 9, 6, 1]
[4] [0, 64, 48, 12, 1]
[5] [0, 625, 500, 150, 20, 1]
[6] [0, 7776, 6480, 2160, 360, 30, 1]
[7] [0, 117649, 100842, 36015, 6860, 735, 42, 1]
[8] [0, 2097152, 1835008, 688128, 143360, 17920, 1344, 56, 1]
"""
@cache
def abel(n: int) -> list[int]:
if n == 0:
return [1]
b = binomial(n - 1)
return [b[k - 1] * n ** (n - k) if k > 0 else 0 for k in range(n + 1)]
@MakeTriangle(abel, "Abel", ["A137452", "A061356", "A139526"], True)
def Abel(n: int, k: int) -> int:
return abel(n)[k]
if __name__ == "__main__":
from _tabltest import TablTest
TablTest(Abel, short=True)
'''
* Statistic about Abel:
The number of ...
all hashes is 199.
distinct hashes is 116.
core triangles is 1.
distinct types is 5.
missing sequences is 86.
all A-numbers is 113.
distinct A-numbers is 58.
The traits of the Abel triangle as represented in the OEIS.
| | A-number| trait | A-name |
|----|---------|------------------|-------------------------------------------------------------------------|
| 1 | A000027 | Inv-DiagCol1 | The positive integers. Also called the natural numbers, the whole numb |
| 2 | A000169 | Std-RowMax | Number of labeled rooted trees with n nodes: n^(n-1) |
| 3 | A000248 | Inv-AltSum | Expansion of e.g.f. exp(x*exp(x)) |
| 4 | A001477 | Std-PolyRow1 | The nonnegative integers |
| 5 | A002378 | Std-DiagRow1 | Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1) |
| 6 | A003725 | Inv-RowSum | E.g.f.: exp( x * exp(-x) ) |
| 7 | A005408 | Rev-PolyRow2 | The odd numbers: a(n) = 2*n + 1 |
| 8 | A005563 | Std-PolyRow2 | a(n) = n*(n+2) = (n+1)^2 - 1 |
| 9 | A007334 | Std-PolyCol2 | Number of spanning trees in the graph K_{n}/e, which results from cont |
| 10 | A009121 | Inv-EvenSum | Expansion of e.g.f. cosh(exp(x)*x) |
| 11 | A009565 | Inv-OddSum | Expansion of e.g.f. sinh(exp(x)*x) |
| 12 | A014026 | Alt-PolyDiag | Inverse of 17th cyclotomic polynomial |
| 13 | A016778 | Rev-PolyRow3 | a(n) = (3*n+1)^2 |
| 14 | A028310 | Std-RowGcd | Expansion of (1 - x + x^2) / (1 - x)^2 in powers of x |
| 15 | A033683 | Alt-TransNat0 | a(n) = 1 if n is an odd square not divisible by 3, otherwise 0 |
| 16 | A036216 | Inv-DiagCol3 | Expansion of 1/(1 - 3*x)^4; 4-fold convolution of A000244 (powers of 3 |
| 17 | A052750 | Std-PosHalf | a(n) = (2*n + 1)^(n - 1) |
| 18 | A052752 | Rev-PolyCol3 | a(n) = (3*n+1)^(n-1) |
| 19 | A059297 | Std-Inv | Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1 |
| 20 | A059299 | Std-RevInv | Triangle of idempotent numbers (version 3), T(n, k) = binomial(n, k) * |
| 21 | A060747 | Inv:Rev-PolyRow2 | a(n) = 2*n - 1 |
| 22 | A067998 | Alt-PolyRow2 | a(n) = n^2 - 2*n |
| 23 | A085527 | Std-NegHalf | a(n) = (2n+1)^n |
| 24 | A089946 | Std-TransNat0 | Secondary diagonal of array A089944, in which the n-th row is the n-th |
| 25 | A137452 | Std-Triangle | Triangular array of the coefficients of the sequence of Abel polynomia |
| 26 | A177885 | Std-AltSum | a(n) = (1-n)^(n-1) |
| 27 | A193678 | Std-PolyDiag | Discriminant of Chebyshev C-polynomials |
| 28 | A195136 | Std-OddSum | a(n) = ((n+1)^(n-1) + (n-1)^(n-1))/2 for n>=1 |
| 29 | A195509 | Inv:Rev-EvenSum | Expansion of e.g.f. (exp(x*exp(x)) + exp(x/exp(x)))/2 |
| 30 | A208879 | Inv:Rev-Poly | Number of words A(n,k), either empty or beginning with the first lette |
| 31 | A216689 | Inv-NegHalf | E.g.f. exp( x * exp(x)^2 ) |
| 32 | A225497 | Std-TransSqrs | Total number of rooted labeled trees over all forests on {1,2,...,n} i |
| 33 | A232006 | Std-Poly | Triangular array read by rows: T(n,k) is the number of simple labeled |
| 34 | A274278 | Std-EvenSum | a(n) = ((n+1)^(n-1) - (n-1)^(n-1))/2 for n>=1 |
| 35 | A274741 | Rev-Poly | Table of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_ |
| 36 | A275707 | Inv:Rev-NegHalf | Number of partial functions f:{1,2,...,n}->{1,2,...,n} such that ever |
| 37 | A320258 | Inv:Rev-PolyDiag | a(n) = n! * [x^n] exp(x*exp(-n*x)) |
| 38 | A356819 | Inv-PosHalf | Expansion of e.g.f. exp(-x * exp(2*x)) |
| 39 | A356820 | Inv:Rev-PolyCol3 | Expansion of e.g.f. exp(-x * exp(3*x)) |
| 40 | A360814 | Inv-DiagSum | Expansion of Sum_{k>=0} x^(2*k) / (1 - k*x)^(k+1) |
| 41 | A362354 | Std-PolyCol3 | a(n) = 3*(n+3)^(n-1) |
| 42 | A367254 | Std-CentralE | a(n) = binomial(2*n - 1, n - 1)*(2*n)^n |
| 43 | A367255 | Std-AccRevSum | a(n) = (n + 1)^(n - 2)*(3*n + 1) |
| 44 | A367256 | Std-BinConv | a(n) = Sum_{k=0..n} binomial(n, k) * binomial(n - 1, k - 1) * n^(n - k |
| 45 | A367257 | Std-InvBinConv | a(n) = Sum_{k=0..n} binomial(n, k) * binomial(n - 1, n - k - 1) * (-n) |
| 46 | A367271 | Inv-CentralE | a(n) = binomial(2*n, n) * n^n |
| 47 | A367272 | Inv-InvBinConv | a(n) = Sum_{k=0..n} binomial(n, k)^2 * k^(n - k) |
| 48 | A367273 | Inv-BinConv | a(n) = Sum_{k=0..n} binomial(n, k)^2 * (k - n)^k |
| 49 | A367274 | Inv:Rev-ColMiddle| a(n) = binomial(n, k) * (n - k)^k where k = floor(n/2) |
| 50 | B000272 | Std-RowSum | Number of trees on n labeled nodes: n^(n-2) with a(0)=1 |
| 51 | B000312 | Alt-AccRevSum | a(n) = n^n; number of labeled mappings from n points to themselves (en |
| 52 | B001788 | Inv-DiagCol2 | a(n) = n*(n+1)*2^(n-2) |
| 53 | B053506 | Std-DiagCol2 | a(n) = (n-1)*n^(n-2) |
| 54 | B053507 | Std-DiagCol3 | a(n) = binomial(n-1,2)*n^(n-3) |
| 55 | B055541 | Alt-TransSqrs | Total number of leaves (nodes of vertex degree 1) in all labeled trees |
| 56 | B065513 | Rev-TransNat0 | Number of endofunctions of [n] with a cycle a->b->c->a and for all x i |
| 57 | B100536 | Inv:Rev-PolyRow3 | a(n) = 3*n^2 - 2 |
| 58 | B366151 | Alt-PolyRow3 | a(n) = T(n, 3), where T(n, k) = Sum_{i=0..n} i^k * binomial(n, i) * (1 |
With much better navigation and the missing sequences:
https://luschny.de/math/oeis/Abel.html
'''