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A276743
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G.f.: Sum_{n>=0} [ Sum_{k>=1} k^n * x^k ]^n.
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4
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1, 1, 3, 12, 63, 447, 4286, 56185, 1008317, 24917676, 849963761, 40142633815, 2633061525012, 240207555735097, 30578843349537575, 5434894746337720676, 1352812180415380719387, 471689727423751377883607, 230943183470327388401886858, 158839247095790148049487792081, 153694547774391577758847456894905
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OFFSET
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0,3
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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 / (1-x)^(n*(n+1)), where A008292 are the Eulerian numbers.
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 63*x^4 + 447*x^5 + 4286*x^6 + 56185*x^7 + 1008317*x^8 + 24917676*x^9 + 849963761*x^10 +...
such that
A(x) = Sum_{n>=0} (x + 2^n*x^2 + 3^n*x^3 +...+ k^n*x^k +...)^n.
Explicitly,
A(x) = 1 + (x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 +...) +
(x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 +...)^2 +
(x + 8*x^2 + 27*x^3 + 64*x^4 + 125*x^5 + 216*x^6 +...)^3 +
(x + 16*x^2 + 81*x^3 + 256*x^4 + 625*x^5 + 1296*x^6 +...)^4 +
(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/(1-x)^2 + (x + x^2)^2/(1-x)^6 + (x + 4*x^2 + x^3)^3/(1-x)^12 + (x + 11*x^2 + 11*x^3 + x^4)^4/(1-x)^20 + (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)^5/(1-x)^30 + (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)^6/(1-x)^42 +...+ [ Sum_{k=1..n} A008292(n,k) * x^k ]^n / (1-x)^(n*(n+1)) +...
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PROG
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(PARI) {a(n) = my(A=1);
A = sum(m=0, n+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) /* Using Eulerian numbers A008292 */
{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/(1-x +Oxn)^(m+1) )^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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