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

%I A033935 #44 Jul 02 2025 16:01:56
%S A033935 1,1,6,93,2716,127905,8848236,844691407,106391894904,17091486402849,
%T A033935 3410496772665940,827540233598615691,239946160014513220896,
%U A033935 81932406267721802925925,32541656017173091541743368,14874686717916861528415671285,7753005946480818323895940923376
%N A033935 Sum of squares of coefficients in full expansion of (z1+z2+...+zn)^n.
%C A033935 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
%H A033935 Alois P. Heinz, <a href="/A033935/b033935.txt">Table of n, a(n) for n = 0..237</a>
%F A033935 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
%F A033935 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
%F A033935 a(n) = n!^2 * [z^n] hypergeom([], [1], z)^n. - _Peter Luschny_, May 31 2017
%p A033935 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A033935       add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
%p A033935     end:
%p A033935 a:= n-> b(n$2):
%p A033935 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 21 2014
%p A033935 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
%t A033935 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 *)
%t A033935 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 *)
%t A033935 Table[SeriesCoefficient[HypergeometricPFQ[{},{1},x]^n, {x,0,n}] n!^2, {n,0,16}] (* _Peter Luschny_, May 31 2017 *)
%Y A033935 Column k=2 of A245397.
%Y A033935 Main diagonal of A287316.
%Y A033935 Cf. A364116.
%K A033935 nonn,easy
%O A033935 0,3
%A A033935 _Warren D. Smith_, Dec 11 1999
%E A033935 More terms from _James Sellers_, Jun 01 2000 and _Vladeta Jovovic_, Jun 05 2000
%E A033935 a(0)=1 inserted by _Alois P. Heinz_, Jul 21 2014