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

0,8

Comments

A287314 provide polynomials and A287315 rational functions generating the columns of the array.

Links

Jeremy Tan, Antidiagonals n = 0..50, flattened

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. B. Richmond and J. 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.

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

Example

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, ...
k=9| 1, 9, 153,   3681,  111321, 3965409, 159700401, 7071121017, ...
       A000384,A169711, A169712, A169713,                            A033935

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}]

Prog

(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 */
(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))
N = 50
i = 0
for n in range(N+1):
    for k in range(n+1):
        print(i, A(k, n-k))
        i += 1
# Jeremy Tan, Dec 10 2021

Crossrefs

Rows: A000007 (k=0), A000012 (k=1), A000984 (k=2), A002893 (k=3), A002895 (k=4), A169714 (k=5), A169715 (k=6).

Columns: A001477(n=1), A000384 (n=2), A169711 (n=3), A169712 (n=4), A169713 (n=5).

Cf. A033935 (diagonal), A287314, A287315, A287318.

Sequence in context: A361950 A183135 A294042 * A322280 A331436 A343097

Adjacent sequences: A287313 A287314 A287315 * A287317 A287318 A287319

Keyword

nonn,tabl

Author

Peter Luschny, May 23 2017

Status

approved