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A248653 E.g.f.: Sum_{n>=0} x^n * (2 + exp(n*x))^n. 4
1, 3, 20, 237, 4276, 107225, 3518526, 145005721, 7285611096, 436297841649, 30590014543930, 2474931380486081, 228308751882636756, 23772216923031342649, 2769853988736186166374, 358463639909150646730665, 51192480930691715108562736, 8021370202848006225125239649 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..17.

FORMULA

E.g.f.: Sum_{n>=0} x^n * exp(n^2*x) / (1 - 2*x*exp(n*x))^(n+1).

EXAMPLE

G.f.: A(x) = 1 + 3*x + 20*x^2/2! + 237*x^3/3! + 4276*x^4/4! + 107225*x^5/5! +...

where the g.f. satisfies following series identity:

A(x) = 1 + x*(2+exp(x)) + x^2*(2+exp(2*x))^2 + x^3*(2+exp(3*x))^3 + x^4*(2+exp(4*x))^4 + x^5*(2+exp(5*x))^5 + x^6*(2+exp(6*x))^6 +...

A(x) = 1/(1-2*x) + x*exp(x)/(1-2*x*exp(x))^2 + x^2*exp(4*x)/(1-2*x*exp(2*x))^3 + x^3*exp(9*x)/(1-2*x*exp(3*x))^4 + x^4*exp(16*x)/(1-2*x*exp(4*x))^5 + x^5*exp(25*x)/(1-2*x*exp(5*x))^6 + x^6*exp(36*x)/(1-2*x*exp(6*x))^7 +...

PROG

(PARI) {a(n, t=2)=local(A=1); A=sum(k=0, n, x^k * (t + exp(k*x +x*O(x^n)))^k); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n, 2), ", "))

(PARI) {a(n, t=2)=local(A=1); A=sum(k=0, n, x^k * exp(k^2*x +x*O(x^n))/(1 - t*x*exp(k*x +x*O(x^n)))^(k+1) ); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n, 2), ", "))

CROSSREFS

Cf. A248615, A193421, A248654.

Sequence in context: A335871 A195135 A201148 * A218374 A219541 A200527

Adjacent sequences: A248650 A248651 A248652 * A248654 A248655 A248656

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 18 2014

STATUS

approved

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Last modified February 4 10:27 EST 2023. Contains 360053 sequences. (Running on oeis4.)