A265943 E.g.f.: Sum_{n>=1} x^(n^2) * exp(3*x^n) / n!.

1, 6, 27, 120, 405, 2538, 5103, 108216, 119529, 8361630, 649539, 1049941764, 6908733, 88290340362, 980827520535, 11464174875312, 731794257, 6190869041758134, 7360989291, 502057591673645340, 28738655044809032421, 30483502110703514466, 721764371007, 330783478823633485162248, 129260083701485621612025, 894884766137985799525518

Offset

1,2

Links

Table of n, a(n) for n=1..26.

Formula

E.g.f.: -exp(3) + Sum_{n>=0} (3 + x^n)^n / n!.

a(n) = Sum_{d|n} 3^(d-n/d) * binomial(d, n/d) * n!/d! for n>=1.

Example

E.g.f.: A(x) = x + 6*x^2/2! + 27*x^3/3! + 120*x^4/4! + 405*x^5/5! + 2538*x^6/6! + ...
where
A(x) = x*exp(3*x) + x^4*exp(3*x^2)/2! + x^9*exp(3*x^3)/3! + x^16*exp(3*x^4)/4! + x^25*exp(3*x^5)/5! + x^36*exp(3*x^6)/6! + ...
also
A(x) = -exp(3) + 1 + (3+x) + (3+x^2)^2/2! + (3+x^3)^3/3! + (3+x^4)^4/4! + (3+x^5)^5/5! + (3+x^6)^6/6! + ...

Mathematica

a[n_] := DivisorSum[n, 3^(#-n/#) * Binomial[#, n/#] * n!/#! &]; Array[a, 25] (* Amiram Eldar, Aug 24 2023 *)

Prog

(PARI) {a(n) = local(A=1); A = sum(m=1, n, x^(m^2) * exp(3*x^m +x*O(x^n)) / m!); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = local(A=1); A = -exp(3) + sum(m=0, n, (3 + x^m +x*O(x^n))^m/m!); if(n==0, 0, n!*polcoeff(A, n))}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = if(n<1, 0, sumdiv(n, d, 3^(d-n/d) * binomial(d, n/d) * n!/d! ) )}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A259209, A259223, A265270.

Sequence in context: A109114 A373302 A080619 * A080620 A080627 A079762

Adjacent sequences: A265940 A265941 A265942 * A265944 A265945 A265946

Keyword

nonn

Author

Paul D. Hanna, Dec 23 2015

Status

approved