A287316 Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 93, 70, 1, 0, 1, 6, 45, 256, 639, 252, 1, 0, 1, 7, 66, 545, 2716, 4653, 924, 1, 0, 1, 8, 91, 996, 7885, 31504, 35169, 3432, 1, 0, 1, 9, 120, 1645, 18306, 127905, 387136, 272835, 12870, 1, 0
Offset: 0
Examples
Arrays start:
k\n| 0 1 2 3 4 5 6 7
---|----------------------------------------------------------------
k=0| 1, 0, 0, 0, 0, 0, 0, 0, ... A000007
k=1| 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
k=2| 1, 2, 6, 20, 70, 252, 924, 3432, ... A000984
k=3| 1, 3, 15, 93, 639, 4653, 35169, 272835, ... A002893
k=4| 1, 4, 28, 256, 2716, 31504, 387136, 4951552, ... A002895
k=5| 1, 5, 45, 545, 7885, 127905, 2241225, 41467725, ... A169714
k=6| 1, 6, 66, 996, 18306, 384156, 8848236, 218040696, ... A169715
k=7| 1, 7, 91, 1645, 36715, 948157, 27210169, 844691407, ...
k=8| 1, 8, 120, 2528, 66424, 2039808, 70283424, 2643158400, ... A385286
k=9| 1, 9, 153, 3681, 111321, 3965409, 159700401, 7071121017, ...
A000384,A169711, A169712, A169713, A033935
Links
- Jeremy Tan, Antidiagonals n = 0..50, flattened
- Nikolai Beluhov, Powers of 2 in High-Dimensional Lattice Walks, arXiv:2506.12789 [math.CO], 2025. See p. 19.
- Ryan S. Bennink, Counting Abelian Squares for a Problem in Quantum Computing, arXiv:2208.02360 [quant-ph], 2022.
- Jonathan M. Borwein, A short walk can be beautiful, preprint, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
- Jonathan M. Borwein and Armin Straub, Mahler measures, short walks and log-sine integrals, preprint, Theoretical Computer Science, Volume 479, 1 April 2013, Pages 4-21.
- Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Some Arithmetic Properties of Short Random Walk Integrals, preprint, FPSAC 2010, San Francisco, USA.
- L. Bruce Richmond and Jeffrey Shallit, Counting Abelian Squares, arXiv:0807.5028 [math.CO], 2008.
- Armin Straub, Arithmetic aspects of random walks and methods in definite integration, Ph. D. Dissertation, School Of Science And Engineering, Tulane University, 2012.
Crossrefs
Programs
-
Maple
A287316_row := proc(k, len) local b, ser; b := k -> BesselI(0, 2*sqrt(x))^k: ser := series(b(k), x, len); seq((i!)^2*coeff(ser,x,i), i=0..len-1) end: for k from 0 to 6 do A287316_row(k, 9) od; A287316_col := proc(n, len) local k, x; sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1): unapply(n!^2*coeff(%, z, n), x); seq(%(j), j=0..len) end: for n from 0 to 7 do A287316_col(n, 9) od;
-
Mathematica
Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (n!)^2, {n, 0, 6}], {k, 0,9}] -
PARI
A287316_row(K, N) = { my(x='x + O('x^(2*N-1))); Vec(serlaplace(serlaplace(substpol(besseli(0,2*x)^K, 'x^2, 'x)))); }; N=8; concat([vector(N, n, n==1)], vector(9, k, A287316_row(k, N))) \\ Gheorghe Coserea, Jan 12 2018
-
PARI
{A(n, k) = if(n<0 || k<0, 0, n!^2 * polcoeff(besseli(0, 2*x + x*O(x^(2*n)))^k, 2*n))}; /* Michael Somos, Dec 30 2021 */ -
PARI
A(k, n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^k, n); \\ Peter Luschny, Jun 24 2025 -
Python
from math import comb from functools import lru_cache @lru_cache(maxsize=None) def A(n,k): if k <= 0: return 0**n return sum(A(i,k-1)*comb(n,i)**2 for i in range(n+1)) for k in range(10): print([A(n, k) for n in range(8)]) # Jeremy Tan, Dec 10 2021
Formula
A(n,k) = A287318(n,k) / binomial(2*n,n).
If a+b=k then A(n,k) = Sum_{i=0..n} A(i,a)*A(n-i,b)*binomial(n,i)^2 (Richmond and Shallit). In particular A(n,k) = Sum_{i=0..n} A(i,k-1)*binomial(n,i)^2. - Jeremy Tan, Dec 10 2021
Comments