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A202990 E.g.f: Sum_{n>=0} 3^n * 2^(n^2) * exp(-2*2^n*x) * x^n/n!. 1
1, 4, 100, 10648, 4477456, 7339040224, 47045881000000, 1186980379913527168, 118530511097526559703296, 47035767668340696232372862464, 74367598058372171073462490000000000, 469253945833810205185008441288962454059008 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.

O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

LINKS

Table of n, a(n) for n=0..11.

FORMULA

a(n) = (3*2^n - 2)^n.

O.g.f.: Sum_{n>=0} 3^n * 2^(n^2) * x^n/(1 + 2*2^n*x)^(n+1).

EXAMPLE

E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 10648*x^3/3! + 4477456*x^4/4! +..

By the series identity, the e.g.f.:

A(x) = exp(-2*x) + 3*2*exp(-2*2*x)*x + 3^2*2^4*exp(-2*2^2*x)*x^2/2! + 3^3*2^9*exp(-2*2^3*x)*x^3/3! +...

expands into:

A(x) = 1 + 4*x + 10^2*x^2/2! + 22^3*x^3/3! + 46^4*x^4/4! + 94^5*x^5/5! +...+ (3*2^n-2)^n*x^n/n! +...

PROG

(PARI) {a(n, q=2, m=3, b=-2)=(m*q^n + b)^n}

(PARI) {a(n, q=2, m=3, b=-2)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}

(PARI) {a(n, q=2, m=3, b=-2)=polcoeff(sum(k=0, n, m^k*q^(k^2)*x^k/(1-b*q^k*x+x*O(x^n))^(k+1)), n)}

CROSSREFS

Cf. A180602, A165327, A202989, A060613, A055601.

Sequence in context: A249939 A224018 A198278 * A324096 A202989 A343183

Adjacent sequences:  A202987 A202988 A202989 * A202991 A202992 A202993

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 26 2011

STATUS

approved

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Last modified September 26 20:57 EDT 2022. Contains 357050 sequences. (Running on oeis4.)