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A276747
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G.f.: Sum_{n>=0} (1-x)^(n*(n+1)) * [ Sum_{k>=1} k^n * x^k ]^n.
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2
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1, 1, 1, 3, 14, 96, 989, 14264, 293081, 8291372, 326486284, 17606371379, 1311003529532, 133789640100606, 18842361596022104, 3651812223033372781, 979595054829206809506, 363619011980801177687068, 187594865162514096249150130, 134684579087971548803896902904, 134956937109764143572996094860839, 189135846049140695927044178145555683, 371258683769470709816610430835777163052
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} [ Sum_{k=1..n} A008292(n,k) * x^k ]^n, where A008292 are the Eulerian numbers.
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 96*x^5 + 989*x^6 + 14264*x^7 + 293081*x^8 + 8291372*x^9 + 326486284*x^10 + 17606371379*x^11 +...
such that
A(x) = Sum_{n>=0} (1-x)^(n*(n+1)) * (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n.
Explicitly,
A(x) = 1 + (1-x)^2 * (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +...) +
(1-x)^6 * (x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 +...)^2 +
(1-x)^12 * (x + 8*x^2 + 27*x^3 + 64*x^4 + 125*x^5 + 216*x^6 +...)^3 +
(1-x)^20 * (x + 16*x^2 + 81*x^3 + 256*x^4 + 625*x^5 + 1296*x^6 +...)^4 +
(1-x)^30 * (x + 32*x^2 + 243*x^3 + 1024*x^4 + 3125*x^5 + 7776*x^6 +...)^5 +
...
The g.f. can be written using the Eulerian numbers like so:
A(x) = 1 + x + (x + x^2)^2 + (x + 4*x^2 + x^3)^3 + (x + 11*x^2 + 11*x^3 + x^4)^4 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6 + (x + 120*x^2 + 1191*x^3 + 2416*x^4 + 1191*x^5 + 120*x^6 + x^7)^7 + (x + 247*x^2 + 4293*x^3 + 15619*x^4 + 15619*x^5 + 4293*x^6 + 247*x^7 + x^8)^8 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n +...
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PROG
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(PARI) {a(n) = my(A=1); A = sum(m=0, n+1, (1-x +x*O(x^n))^(m*(m+1)) * sum(k=1, n+1, k^m*x^k +x*O(x^n))^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = sum(m=0, n+1, sum(k=1, m, A008292(m, k)*x^k +Oxn )^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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