%I #13 Mar 21 2024 04:59:19

%S 1,6,93,2448,92505,4589568,283008621,20903023872,1800986581521,

%T 177455695795200,19690717755237309,2430478269127673856,

%U 330392930155527272553,49053029845102480576512,7898602773992589665290125,1371137549213022697047785472,255275516636592894833768588961

%N a(n) = coefficient of x^n*y^(n+1)/n! in (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ).

%H Paul D. Hanna, <a href="/A359928/b359928.txt">Table of n, a(n) for n = 1..200</a>

%F E.g.f. A(x) = Sum_{n>=1} a(n) * x^n/n! may be defined by the following.

%F (1) a(n) = [x^n*y^(n+1)/n!] (1/2)*log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ).

%F (2) A(x) = lim_{N->oo} (1/N)*log( Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ).

%F a(n) ~ c * d^n * n! / n^(5/2), where d = 12.7029497597456784744445675253711147535742245945208995646083627... and c = 0.15440395598650604464793307483290467035754174771895993579108... - _Vaclav Kotesovec_, Mar 21 2024

%e E.g.f.: A(x) = x + 6*x^2/2! + 93*x^3/3! + 2448*x^4/4! + 92505*x^5/5! + 4589568*x^6/6! + 283008621*x^7/7! + 20903023872*x^8/8! + 1800986581521*x^9/9! + 177455695795200*x^10/10! + ...

%e Exponentiation yields the e.g.f. of A319147:

%e exp(A(x)) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2965*x^4/4! + 111856*x^5/5! + 5528419*x^6/6! + 339433984*x^7/7! + 24965493865*x^8/8! + 2142654088960*x^9/9! + ... + A319147(n)*x^n/n! + ...

%e which equals the limit

%e exp(A(x)) = lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

%e RELATED SEQUENCES.

%e a(n) is divisible by n where a(n)/n begins:

%e [1, 3, 31, 612, 18501, 764928, 40429803, 2612877984, 200109620169, ...].

%o (PARI) {a(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^m*(m + 2*y)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A359927, A319834, A318634.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 20 2023