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A326011
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a(n) = (n+1) * (2^n + 1)^n.
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2
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1, 6, 75, 2916, 417605, 234812358, 527932234375, 4755738419928072, 171280331996409907209, 24606864966197875457438730, 14080929986159936046600341796875, 32073236633246852578917758577924120588, 290760173774986242601808360162358149769707533, 10492680499171055486742235424276666079725581443186702, 1507792223578968167717594884445653164343553232898773193359375
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OFFSET
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0,2
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COMMENTS
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More generally, the following sums are equal:
(1) Sum_{n>=0} binomial(n+k-1, n) * y^n * (F + G^n)^n,
(2) Sum_{n>=0} binomial(n+k-1, n) * y^n * G^(n^2) / (1 - y*F*G^n)^(n+k),
for any fixed integer k; here, k = 2 and y = x, F = 1, G = 2.
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LINKS
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Table of n, a(n) for n=0..14.
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FORMULA
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O.g.f.: Sum_{n>=0} (n+1) * (2^n + 1)^n * x^n.
O.g.f.: Sum_{n>=0} (n+1) * 2^(n^2) * x^n / (1 - 2^n*x)^(n+2).
E.g.f.: sum_{n>=0} (n+1 + 2^n*x) * 2^(n^2) * exp(2^n*x) * x^n/n!.
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EXAMPLE
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O.g.f.: A(x) = 1 + 6*x + 75*x^2 + 2916*x^3 + 417605*x^4 + 234812358*x^5 + 527932234375*x^6 + 4755738419928072*x^7 + ... + (n+1)*(2^n + 1)^n*x^n + ...
such that
A(x) = 1/(1 - x)^2 + 2*2*x/(1 - 2*x)^3 + 3*2^4*x^2/(1 - 2^2*x)^4 + 4*2^9*x^3/(1 - 2^3*x)^5 + 5*2^16*x^4/(1 - 2^4*x)^6 + 6*2^25*x^5/(1 - 2^5*x)^7 + 7*2^36*x^6/(1 - 2^6*x)^8 + ... + (n+1)*2^(n^2)*x^n/(1 - 2^n*x)^(n+2) + ...
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PROG
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(PARI) {a(n) = (n+1) * (2^n + 1)^n}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* O.g.f. */
{a(n) = my(A = sum(m=0, n, (m+1) * 2^(m^2) * x^m / (1 - 2^m*x +x*O(x^n))^(m+2) )); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* E.g.f. */
{a(n) = my(A = sum(m=0, n, (m+1 + 2^m*x) * 2^(m^2) * exp(2^m*x +x*O(x^n)) * x^m/m! )); n!*polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
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CROSSREFS
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Cf. A136516, A326012.
Sequence in context: A126462 A081066 A185289 * A263228 A229571 A016090
Adjacent sequences: A326008 A326009 A326010 * A326012 A326013 A326014
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jun 05 2019
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STATUS
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approved
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