A033935 -id:A033935 - cp's OEIS frontend

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

Peter Luschny, May 23 2017

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

			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

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.

Rows: A000007 (k=0), A000012 (k=1), A000984 (k=2), A002893 (k=3), A002895 (k=4), A169714 (k=5), A169715 (k=6), A385286 (k=8).
Columns: A001477(n=1), A000384 (n=2), A169711 (n=3), A169712 (n=4), A169713 (n=5).
Cf. A033935 (diagonal), A287314, A287315, A287318.

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;

Table[Table[SeriesCoefficient[BesselI[0, 2 Sqrt[x]]^k, {x, 0, n}] (n!)^2, {n, 0, 6}], {k, 0,9}]

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

{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 */

A(k, n) = my(x='x+O('x^(n+1))); n!^2*polcoeff(hypergeom([], [1], x)^k, n); \\ Peter Luschny, Jun 24 2025

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

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

A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 10, 1, 1, 6, 27, 35, 1, 1, 10, 93, 256, 126, 1, 1, 18, 381, 2716, 3125, 462, 1, 1, 34, 1785, 36628, 127905, 46656, 1716, 1, 1, 66, 9237, 591460, 7120505, 8848236, 823543, 6435, 1, 1, 130, 51033, 11007556, 495872505, 2443835736, 844691407, 16777216, 24310

Offset: 0

Alois P. Heinz, Jul 21 2014

			A(3,2) = 93: (z1+z2+z3)^3 = z1^3 +3*z1^2*z2 +3*z1^2*z3 +3*z1*z2^2 +6*z1*z2*z3 +3*z1*z3^2 +z2^3 +3*z2^2*z3 +3*z2*z3^2 +z3^3 => 1^2+3^2+3^2+3^2+6^2+3^2+1^2+3^2+3^2+1^2 = 93.
Square array A(n,k) begins:
0 :    1,    1,      1,       1,         1,           1, ...
1 :    1,    1,      1,       1,         1,           1, ...
2 :    3,    4,      6,      10,        18,          34, ...
3 :   10,   27,     93,     381,      1785,        9237, ...
4 :   35,  256,   2716,   36628,    591460,    11007556, ...
5 :  126, 3125, 127905, 7120505, 495872505, 41262262505, ...

Alois P. Heinz, Antidiagonals n = 0..50, flattened

Columns k=0-10 give: A001700(n-1) for n>0, A000312, A033935, A055733, A055740, A246240, A246241, A246242, A246243, A246244, A246245.
Rows n=0+1, 2 give: A000012, A052548.
Main diagonal gives A245398.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] * Binomial[n, j]^(k-1)/j!, {j, 0, n}]]]; A[n_, k_] := n!*b[n, n, k]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.

A364116 a(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^n for n >= 0.

Original entry on oeis.org

1, 3, 73, 5623, 908001, 251831261, 106898093065, 64439674636863, 52344140654486017, 55113399257643294769, 73004404532578627776801, 118810038754810358401521065, 233027150139808176596750408337, 542098915811219991386976197616441

Offset: 0

Peter Bala, Jul 08 2023

Main diagonal of A364113.
Compare with the two types of Apéry numbers A005258 and A005259, which are related to the Legendre polynomials by A005258(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x)) and A005259(n) = [x^k] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^2.
A005258 is the main diagonal of A108625 and A005259 is the main diagonal of A143007.

Cf. A005258, A005259, A108625, A143007, A364113, A364114, A364115, A364117, A364301.

a(n) := coeff(series( 1/(1-x)* LegendreP(n,(1+x)/(1-x))^n, x, 21), x, n):
seq(a(n), n = 0..20);

Table[SeriesCoefficient[1/(1 - x) * LegendreP[n, (1 + x)/(1 - x)]^n, {x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Jul 09 2023 *)

Conjectures:
1) a(p) == 2*p + 1 (mod p^4) for all primes p >= 3 (checked up to p = 101).
More generally, the supercongruence a(p^k) == 2*p^k + 1 (mod p^(3+k)) may hold for all primes p >= 5 and all k >= 1.
2) a(p-1) == 1 (mod p^3) for all primes p >= 5 (checked up to p = 101).
More generally, the supercongruence a(p^k - p^(k-1)) == 1 (mod p^(2+k)) may hold for all primes p >= 5 and all k >= 1.
From Vaclav Kotesovec, Jul 10 2023: (Start)
a(n) ~ c * d^n * n^(2*n - 1/2), where d = 2.102423770105721036432437141524634595160013830317976222331887376263238499... (the same as for A033935) and c = 1.325068544739430738025458046917491360304162175529817456184402029433873399...
a(n) ~ A033935(n) * exp(2*n + 1) / (2*Pi*n).
a(n) ~ A033935(n) * exp(1) * n^(2*n) / n!^2. (End)

A055733 Sum of third powers of coefficients in full expansion of (z1+z2+...+zn)^n.

Original entry on oeis.org

1, 1, 10, 381, 36628, 7120505, 2443835736, 1351396969615, 1127288317316008, 1349611750487720817, 2230372438317527996620, 4930842713588476723120511, 14211567663513739084746570600, 52259895270824126097423028107277, 240736564755509319272061470644316416

Offset: 0

Vladeta Jovovic, Jun 09 2000

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A033935.

Column k=3 of A245397.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^2/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014
A055733 := proc(n) series(hypergeom([],[1,1],z)^n,z=0,n+1): n!^3*coeff(%,z,n) end: seq(A055733(n), n=0..14); # Peter Luschny, May 31 2017

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1]*Binomial[n, j]^2 / j!, {j, 0, n}]]]; a[n_] := n!*b[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)
Table[SeriesCoefficient[HypergeometricPFQ[{}, {1,1}, x]^n, {x,0,n}] n!^3, {n,0,14}] (* Peter Luschny, May 31 2017 *)

a(n) is coefficient of x^n in expansion of n!^3*(1+x/1!^3+x^2/2!^3+x^3/3!^3+...+x^n/n!^3)^n.
a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 1.74218173246413..., c = 0.5728782413434... . - Vaclav Kotesovec, Aug 20 2014
a(n) = (n!)^3 * [z^n] hypergeom([], [1, 1], z)^n. - Peter Luschny, May 31 2017

a(0)=1 inserted by Alois P. Heinz, Jul 21 2014

A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.

Original entry on oeis.org

1, 2, 36, 1860, 190120, 32232060, 8175770064, 2898980908824, 1369263687414480, 830988068906518380, 630109741730668410640, 583773362067938664133512, 648851848280206013365243776, 852146184628067383511375555000, 1305460597778526044143501996708800, 2307324514460203126471248458864413200

Offset: 0

Ilya Gutkovskiy, Apr 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..212

Main diagonal of A287318.

Cf. A000984, A002894, A002896, A033935, A039699, A287317, A361297.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> (2*n)!*b(n$2)/n!^2:
seq(a(n), n=0..17);  # Alois P. Heinz, Jan 29 2023

Table[(2 n)! SeriesCoefficient[BesselI[0, 2 x]^n, {x, 0, 2 n}], {n, 0, 15}]

a(n) = A287318(n,n).
a(n) ~ c * d^n * n^(2*n), where c = 1.72802011936236389522137050964080... and d = 1.1381284656425793765251319541847869000364101065484286935... - Vaclav Kotesovec, Apr 26 2018
a(n) = A000984(n)*A033935(n). - Alois P. Heinz, Jan 30 2023

A336665 a(n) = (n!)^2 * [x^n] 1 / BesselJ(0,2*sqrt(x))^n.

Original entry on oeis.org

1, 1, 10, 255, 12196, 939155, 106161756, 16554165495, 3404986723720, 893137635015219, 290965846152033460, 115256679181251696803, 54552992572663333862400, 30406695393635479756804525, 19712738332895648545008815416, 14707436666152282009334357074335

Offset: 0

Ilya Gutkovskiy, Jul 29 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..99

Cf. A000275, A033935, A336271, A336638, A336639.

Main diagonal of A340986.

Table[(n!)^2 SeriesCoefficient[1/BesselJ[0, 2 Sqrt[x]]^n, {x, 0, n}], {n, 0, 15}]
A287316[n_, k_] := A287316[n, k] = If[n == 0, 1, If[k < 1, 0, Sum[Binomial[n, j]^2 A287316[n - j, k - 1], {j, 0, n}]]]; b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[(-1)^(j + 1) Binomial[n, j]^2 A287316[j, k] b[n - j, k], {j, 1, n}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 15}]

a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 3.431031961004073074179854315227049823720211... and c = 0.31156343453490677011135864540173577785263... - Vaclav Kotesovec, May 04 2024

A055740 Sum of fourth powers of coefficients in full expansion of (z1+z2+...+zn)^n.

Original entry on oeis.org

1, 1, 18, 1785, 591460, 495872505, 882463317636, 2956241639184631, 17088644286346128840, 159584255348964655673745, 2286523844910576580400966980, 48220116744252542032928364578451, 1446485887751234540636003724054342864, 59981372975740557234356339667492583487125

Offset: 0

Vladeta Jovovic, Jun 09 2000

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A033935.

Column k=4 of A245397.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)^3/j!, j=0..n)))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1]*Binomial[n, j]^3 /j!, {j, 0, n}]]]; a[n_] := n!*b[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

a(n) is coefficient of x^n in expansion of n!^4*(1+x/1!^4+x^2/2!^4+x^3/3!^4+...+x^n/n!^4)^n.

a(n) ~ c * d^n * (n!)^4 / sqrt(n), where d = 1.511958716403..., c = 0.6632048858... . - Vaclav Kotesovec, Aug 20 2014

a(0)=1 inserted by Alois P. Heinz, Jul 21 2014

cp's OEIS Frontend

A287316 Square array A(n,k) = (n!)^2 [x^n] BesselI(0, 2*sqrt(x))^k read by antidiagonals.

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A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A364116 a(n) = [x^n] 1/(1 - x) * Legendre_P(n, (1 + x)/(1 - x))^n for n >= 0.

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A055733 Sum of third powers of coefficients in full expansion of (z1+z2+...+zn)^n.

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A303503 a(n) = (2*n)! * [x^(2*n)] BesselI(0,2*x)^n.

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A336665 a(n) = (n!)^2 * [x^n] 1 / BesselJ(0,2*sqrt(x))^n.

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A055740 Sum of fourth powers of coefficients in full expansion of (z1+z2+...+zn)^n.

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A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.