Revision History for A218669
(Underlined text is an addition;
strikethrough text is a deletion.)
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#4 by Paul D. Hanna at Sun Nov 04 15:05:55 EST 2012
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#3 by Paul D. Hanna at Sun Nov 04 15:04:42 EST 2012
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| NAME
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O.g.f.: A(x) = Sum_{n>=0} 1/(1-n^3*x)^n * x^n/n! * exp(-x/(1-n^3*x)).
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| COMMENTS
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Compare g.f. to the curious identity:
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| EXAMPLE
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O.g.f.: A(x) = 1 + x^2 + 7*x^3 + 97*x^4 + 1561*x^5 + 41136*x^6 +...
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| PROG
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((PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A=sum(k=0, n, 1/(1-k^3*X)^k*x^k/k!*exp(-X/(1-k^3*X))); polcoeff(A, n)}
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| CROSSREFS
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Cf. A218667, A218668, A218670, A217900.
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#2 by Paul D. Hanna at Sun Nov 04 14:06:48 EST 2012
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allocated for Paul D. Hanna
O.g.f.: A(x) = Sum_{n>=0} 1/(1-n^3*x)^n * x^n/n! * exp(-x/(1-n^3*x)).
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| DATA
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1, 0, 1, 7, 97, 1561, 41136, 1551814, 72440460, 4281320257, 324623105584, 30086950057627, 3299720918091511, 428431079916572044, 65637957066642609845, 11659659637028895337265, 2367270866164121777222596, 546795407830461739380895161, 143176487805296033192642234802
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| OFFSET
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0,4
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| COMMENTS
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Compare g.f. to the curious identity:
1/(1-x^2) = Sum_{n>=0} (1+n*x)^n * x^n/n! * exp(-x*(1+n*x)).
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| EXAMPLE
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O.g.f.: A(x) = 1 + x^2 + 7*x^3 + 97*x^4 + 1561*x^5 + 41136*x^6 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-8*x)^2/2!*exp(-x/(1-8*x)) + x^3/(1-27*x)^3/3!*exp(-x/(1-27*x)) + x^4/(1-64*x)^4/4!*exp(-x/(1-64*x)) + x^5/(1-125*x)^5/5!*exp(-x/(1-125*x)) +...
simplifies to a power series in x with integer coefficients.
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| PROG
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(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A=sum(k=0, n, 1/(1-k^3*X)^k*x^k/k!*exp(-X/(1-k^3*X))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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| CROSSREFS
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Cf. A218667, A218668, A218670, A217900.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna, Nov 04 2012
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| STATUS
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approved
editing
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#1 by Paul D. Hanna at Sun Nov 04 06:16:52 EST 2012
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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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