Revision History for A244822
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#9 by Vaclav Kotesovec at Fri Jul 11 07:44:11 EDT 2014
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#8 by Vaclav Kotesovec at Fri Jul 11 07:44:06 EDT 2014
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#7 by Vaclav Kotesovec at Fri Jul 11 07:21:13 EDT 2014
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#6 by Vaclav Kotesovec at Fri Jul 11 07:21:08 EDT 2014
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| MATHEMATICA
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Flatten[{1, Table[Sum[Binomial[n, k]*k^(n-k)*4^(k*(n-k)), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 11 2014 *)
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| STATUS
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approved
editing
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#5 by Vaclav Kotesovec at Fri Jul 11 03:29:29 EDT 2014
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#4 by Vaclav Kotesovec at Fri Jul 11 03:25:46 EDT 2014
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| NAME
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E.g.f.: Sum_{n>=0} exp(n*4^n*x) * x^n/n!.
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| LINKS
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Vaclav Kotesovec, <a href="http://oeis.org/A244820/a244820.pdf">Asymptotic of sequences A244820, A244821 and A244822</a>
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| FORMULA
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O.g.f.: Sum_{n>=0} x^n/(1 - n*4^n*x)^(n+1).
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| EXAMPLE
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E.g.f.: A(x) = 1 + x + 9*x^2/2! + 145*x^3/3! + 7169*x^4/4! + 702721*x^5/5! +...
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| PROG
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((PARI) {a(n) = sum(k=0, n, binomial(n, k) * k^(n-k) * 4^(k*(n-k)) )}
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| CROSSREFS
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Cf. A244820, A244821.
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| STATUS
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approved
editing
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#3 by Paul D. Hanna at Sun Jul 06 13:49:13 EDT 2014
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#2 by Paul D. Hanna at Sun Jul 06 13:49:10 EDT 2014
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| NAME
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allocated for Paul D. Hanna
E.g.f.: Sum_{n>=0} exp(n*4^n*x) * x^n/n!.
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| DATA
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1, 1, 9, 145, 7169, 702721, 173051905, 86399717377, 99140462706689, 233906591488868353, 1206701231035902853121, 12911553576265127971258369, 292981931017250265780757463041, 13856406784814016950200694583853057, 1362697700959059311763086710096185524225
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| OFFSET
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0,3
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| FORMULA
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O.g.f.: Sum_{n>=0} x^n/(1 - n*4^n*x)^(n+1).
a(n) = Sum_{k=0..n} C(n,k) * k^(n-k) * 4^(k*(n-k)).
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| EXAMPLE
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E.g.f.: A(x) = 1 + x + 9*x^2/2! + 145*x^3/3! + 7169*x^4/4! + 702721*x^5/5! +...
where
A(x) = 1 + exp(4*x)*x + exp(4^2*x)^2*x^2/2! + exp(4^3*x)^3*x^3/3! + exp(4^4*x)^4*x^4/4! + exp(4^5*x)^5*x^5/5! + exp(4^6*x)^6*x^6/6! +...
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| PROG
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(PARI) {a(n) = sum(k=0, n, binomial(n, k) * k^(n-k) * 4^(k*(n-k)) )}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, exp(k*4^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*4^k*x +x*O(x^n))^(k+1)), n)}
for(n=0, 25, print1(a(n), ", "))
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| CROSSREFS
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Cf. A244820, A244821.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna, Jul 06 2014
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| STATUS
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approved
editing
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#1 by Paul D. Hanna at Sun Jul 06 13:40:12 EDT 2014
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| NAME
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allocated for Paul D. Hanna
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| KEYWORD
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allocated
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| STATUS
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approved
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