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A306067
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E.g.f. A(x) satisfies: Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + (k+1)*A(x) = 1.
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3
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1, 4, 21, 178, 2279, 39066, 835132, 21400198, 640239525, 21920851282, 845615003996, 36298192983482, 1716348366690487, 88653661788525666, 4967006270867149524, 300043327305644202366, 19440451816128996788777, 1344909407655243937857826, 98949254253416815493778796, 7714902418308597200477578514, 635444724815621395463510504211, 55134789286331454820101232131938
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + k*A(x) = -x.
(2) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + (k+1)*A(x) = 1.
(3) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k+1) + k*A(x) = 1/A(x).
a(n)/n! ~ c * d^n / n^(3/2), where d = 4.423034555284689... and c = 3.17922741818... - Vaclav Kotesovec, Jul 12 2018
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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 21*x^2/2! + 178*x^3/3! + 2279*x^4/4! + 39066*x^5/5! + 835132*x^6/6! + 21400198*x^7/7! + 640239525*x^8/8! + 21920851282*x^9/9! + 845615003996*x^10/10! + ...
such that
1 = (0 + A(x)) - (1 + A(x))*(0 + 2*A(x))*x + (2 + A(x))*(1 + 2*A(x))*(0 + 3*A(x))*x^2/2! - (3 + A(x))*(2 + 2*A(x))*(1 + 3*A(x))*(0 + 4*A(x))*x^3/3! + (4 + A(x))*(3 + 2*A(x))*(2 + 3*A(x))*(1 + 4*A(x))*(0 + 5*A(x))*x^4/4! - (5 + A(x))*(4 + 2*A(x))*(3 + 3*A(x))*(2 + 4*A(x))*(1 + 5*A(x))*(0 + 6*A(x))*x^5/5! + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = -Vec( sum(n=0, #A, (-x)^n/n!* prod(k=0, n, (n-k) + (k+1)*Ser(A) ) ) )[#A] ); n!*A[n+1]}
for(n=0, 20, 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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