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A304641
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E.g.f. A(x) satisfies: 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.
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1
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1, 2, 6, 74, 3078, 228842, 25277286, 3837501194, 762731347398, 191798593122602, 59475206565622566, 22290155840476400714, 9933314218291366691718, 5192540728710234994272362, 3147427468437058629798524646, 2190237887318737512524514442634, 1734606000858253287464231519860038, 1551466530739217915273113571521758122, 1556475858078242120174483544923467343526
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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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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( exp((n+1)*x) - A(x) )^n.
(2) 1 = Sum_{n>=0} exp(n*(n+1)*x) / (1 + exp(n*x)*A(x))^(n+1).
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 74*x^3/3! + 3078*x^4/4! + 228842*x^5/5! + 25277286*x^6/6! + 3837501194*x^7/7! + 762731347398*x^8/8! + 191798593122602*x^9/9! + 59475206565622566*x^10/10! + ...
such that
1 = 1 + (exp(2*x) - A(x)) + (exp(3*x) - A(x))^2 + (exp(4*x) - A(x))^3 + (exp(5*x) - A(x))^4 + (exp(6*x) - A(x))^5 + (exp(7*x) - A(x))^6 + (exp(8*x) - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + exp(2*x)/(1 + exp(x)*A(x))^2 + exp(6*x)/(1 + exp(2*x)*A(x))^3 + exp(12*x)/(1 + exp(3*x)*A(x))^4 + exp(20*x)/(1 + exp(4*x)*A(x))^5 + exp(30*x)/(1 + exp(5*x)*A(x))^6 + exp(42*x)/(1 + exp(6*x)*A(x))^7 + ...
RELATED SERIES.
log(A(x)) = 2*x + 2*x^2/2! + 54*x^3/3! + 2570*x^4/4! + 199590*x^5/5! + 22598762*x^6/6! + 3488755494*x^7/7! + 701959131050*x^8/8! + 178186466260710*x^9/9! + 55669778154059882*x^10/10! + ...
exp(-x) * A(x) = 1 + x + 3*x^2/2! + 61*x^3/3! + 2811*x^4/4! + 214141*x^5/5! + 23949003*x^6/6! + 3665260621*x^7/7! + 732726498171*x^8/8! + 185070066199261*x^9/9! + 57591088296085803*x^10/10! + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, (exp((m+1)*x +x*O(x^#A)) - Ser(A))^m ) )[#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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