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A276746
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G.f.: Sum_{n>=0} exp(-n*x) * [ Sum_{k>=1} k^n * x^k/k! ]^n.
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3
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1, 1, 1, 3, 11, 59, 439, 4472, 62935, 1209430, 32051191, 1166861194, 58721422238, 4088067759572, 394939948283555, 53103347018217191, 9947926726521152127, 2605793744856238449900, 954325686563453398851948, 490307345827423220333318823, 353233139353798743949103145806, 357764578484543977966398255436596, 509302875450935550273793437784438990, 1020636223630190179625684042390508142277
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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} S2(n,k) * x^k ]^n, where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 11*x^4 + 59*x^5 + 439*x^6 + 4472*x^7 + 62935*x^8 + 1209430*x^9 + 32051191*x^10 + 1166861194*x^11 + 58721422238*x^12 +...
such that
A(x) = Sum_{n>=0} exp(-n*x)*(x + 2^n*x^2/2! + 3^n*x^3/3! +...+ k^n*x^k/k! +...)^n.
Explicitly,
A(x) = 1 + exp(-x) * (x + 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 5*x^5/5! +...) +
exp(-2*x) * (x + 4*x^2/2! + 9*x^3/3! + 16*x^4/4! + 25*x^5/5! +...)^2 +
exp(-3*x) * (x + 8*x^2/2! + 27*x^3/3! + 64*x^4/4! + 125*x^5/5! +...)^3 +
exp(-4*x) * (x + 16*x^2/2! + 81*x^3/3! + 256*x^4/4! + 625*x^5/5! +...)^4 +
exp(-5*x) * (x + 32*x^2/2! + 243*x^3/3! + 1024*x^4/4! + 3125*x^5/5! +...)^5 +
...
The g.f. can be written using the Stirling2 numbers like so:
A(x) = 1 + x + (x + x^2)^2 + (x + 3*x^2 + x^3)^3 + (x + 7*x^2 + 6*x^3 + x^4)^4 + (x + 15*x^2 + 25*x^3 + 10*x^4 + x^5)^5 + (x + 31*x^2 + 90*x^3 + 65*x^4 + 15*x^5 + x^6)^6 + (x + 63*x^2 + 301*x^3 + 350*x^4 + 140*x^5 + 21*x^6 + x^7)^7 + (x + 127*x^2 + 966*x^3 + 1701*x^4 + 1050*x^5 + 266*x^6 + 28*x^7 + x^8)^8 +...+ [ Sum_{k=1..n} S2(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, exp(-m*x +x*O(x^n)) * sum(k=1, n+1, k^m*x^k/k! +x*O(x^n))^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n, sum(k=1, m, stirling(m, k, 2)*x^k +x*O(x^n) )^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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