A033935 Sum of squares of coefficients in full expansion of (z1+z2+...+zn)^n.

1, 1, 6, 93, 2716, 127905, 8848236, 844691407, 106391894904, 17091486402849, 3410496772665940, 827540233598615691, 239946160014513220896, 81932406267721802925925, 32541656017173091541743368, 14874686717916861528415671285, 7753005946480818323895940923376

Offset

0,3

Comments

Two samples of size n are taken from an urn containing infinitely many marbles of n distinct colors. a(n)/n^(2*n) is the probability that the two samples match. That is, they contain the same number of each color of marbles without regard to order. - Geoffrey Critzer, Apr 19 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 0..237

Formula

a(n) is coefficient of x^n in expansion of n!^2*(1 + x/1!^2 + x^2/2!^2 + x^3/3!^2 + ... + x^n/n!^2)^n. - Vladeta Jovovic, Jun 09 2000

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.1024237701057210364324371415246345951600138303179762223318873762632384990..., c = 0.487465475752598098146353111500372156824276600165331887960705498284416... - Vaclav Kotesovec, Jul 29 2014, updated Jul 10 2023

a(n) = n!^2 * [z^n] hypergeom([], [1], z)^n. - Peter Luschny, May 31 2017

Maple

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-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2014
A033935:= proc(n) series(hypergeom([], [1], z)^n, z=0, n+1): n!^2*coeff(%, z, n) end: seq(A033935(n), n=0..16); # Peter Luschny, May 31 2017

Mathematica

Table[nn=n; n!^2 Coefficient[Series[(Sum[x^k/k!^2, {k, 0, nn}])^n, {x, 0, nn}], x^n], {n, 1, 20}] (* Geoffrey Critzer, Apr 19 2014 *)
Flatten[{1, Table[n!^2*Coefficient[Series[BesselI[0, 2*Sqrt[x]]^n, {x, 0, n}], x^n], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 29 2014 *)
Table[SeriesCoefficient[HypergeometricPFQ[{}, {1}, x]^n, {x, 0, n}] n!^2, {n, 0, 16}] (* Peter Luschny, May 31 2017 *)

Crossrefs

Column k=2 of A245397.

Main diagonal of A287316.

Cf. A364116.

Sequence in context: A328427 A103212 A359928 * A218682 A078103 A221525

Adjacent sequences: A033932 A033933 A033934 * A033936 A033937 A033938

Keyword

nonn,easy

Author

Warren D. Smith, Dec 11 1999

Extensions

More terms from James A. Sellers, Jun 01 2000 and Vladeta Jovovic, Jun 05 2000

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

Status

approved