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Cf. A244821, A244822, A245076.

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Flatten[{1, Table[Sum[Binomial[n, k]*k^(n-k)*2^(k*(n-k)), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 11 2014 *)

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

Vaclav Kotesovec, <a href="/A244820/a244820.pdf">Asymptotic of sequences A244820, A244821 and A244822</a>

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

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 513*x^4/4! + 11281*x^5/5! +...

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * k^(n-k) * 2^(k*(n-k)) )}

Cf. A244821, A244822.

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allocated for Paul D. Hanna

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

1, 1, 5, 37, 513, 11281, 400513, 22016065, 1861165057, 238780240129, 46058931537921, 13292137309135873, 5694523821282066433, 3612945464580972908545, 3375333122746593847050241, 4635513066684099431721615361, 9320885421210678888076169707521, 27400026186934818737377587727761409

0,3

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

a(n) = Sum_{k=0..n} C(n,k) * k^(n-k) * 2^(k*(n-k)).

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 513*x^4/4! + 11281*x^5/5! +...

where

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

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * k^(n-k) * 2^(k*(n-k)) )}

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

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

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

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

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

Cf. A244821, A244822.

allocated

nonn

Paul D. Hanna, Jul 06 2014

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allocated for Paul D. Hanna

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