A359918 a(n) = coefficient of x^n*y^(n+1)/n! in (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ).

1, 2, 21, 304, 6985, 205056, 7607509, 337188608, 17495079921, 1038495001600, 69496455755221, 5176052539987968, 424783071501394489, 38087843235679268864, 3704990294840345047125, 388631778963216211050496, 43729459820175064700435041, 5254332451028464517449777152

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

a(n) ~ c * d^n * n! / n^(5/2), where d = 7.68892218919697462312... and c = 0.1314019396717313039... - Vaclav Kotesovec, Mar 21 2024

Example

E.g.f.: A(x) = x + 2*x^2/2! + 21*x^3/3! + 304*x^4/4! + 6985*x^5/5! + 205056*x^6/6! + 7607509*x^7/7! + 337188608*x^8/8! + 17495079921*x^9/9! + 1038495001600*x^10/10! + ...
Exponentiation yields the e.g.f. of A359917:
exp(A(x)) = 1 + x + 3*x^2/2! + 28*x^3/3! + 413*x^4/4! + 9216*x^5/5! + 268327*x^6/6! + 9831424*x^7/7! + 432251577*x^8/8! +...+ A359917(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 1, 7, 76, 1397, 34176, 1086787, 42148576, 1943897769, 103849500160, ...].

Prog

(PARI) {a(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + 2*y^2)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A359917, A359926, A359928, A319834, A318634.

Sequence in context: A229036 A097627 A094797 * A234525 A230382 A189489

Adjacent sequences: A359915 A359916 A359917 * A359919 A359920 A359921

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2023

Status

approved