A196560 - OEIS

%I #9 Aug 25 2024 14:01:29

%S 1,1,3,20,205,2624,24793,283522,3639005,50426826,740744940,

%T 10801827249,163698355616,2554965416964,40878247859612,

%U 667841855292388,11051724909284834,185702751266940874,3162454792706586691,54508849210857505845

%N G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4 * x^k]^2 * x^n/n ).

%e G.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 205*x^4 + 2624*x^5 + 24793*x^6 +...

%e where

%e log(A(x)) = (1 + x)^2*x + (1+2^4*x+x^2)^2*x^2/2 + (1+3^4*x+3^4*x^2+x^3)^2*x^3/3 + (1+4^4*x+6^4*x^2+4^4*x^3+x^4)^2*x^4/4 +...

%e More explicitly,

%e log(A(x)) = x + 5*x^2/2 + 52*x^3/3 + 733*x^4/4 + 11926*x^5/5 + 129944*x^6/6 +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^4*x^k)^2*x^m/m)+x*O(x^n)), n)}

%Y Cf. A166898, A180718, A196559.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 03 2011