login
A287316
Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.
12
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
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, ...
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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 23 2017
STATUS
approved