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

Author

Warren D. Smith, Dec 11 1999

Keywords

• nonn
• easy

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

Crossrefs

Column k=2 of A245397.

Main diagonal of A287316.

Cf. A364116.

Programs

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 *)

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

Extensions

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

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