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A338178
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G.f.: A(x) satisfies 1 = Sum_{n>=0} Product_{k=n..2*n-1} ( (1+x)^k - A(x) ).
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3
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1, 1, 2, 23, 383, 8456, 228657, 7268077, 264627570, 10842464809, 493454895161, 24695915073150, 1348165575801780, 79740451015249487, 5080946666286668723, 347046897108675507548, 25300042941361404342404, 1960975025091063811051791, 161045636717926154800016147, 13970534533583992498224467231
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OFFSET
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0,3
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COMMENTS
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Compare the g.f. A(x) to functions B(x) and C(x) defined by
A338181: 1 = Sum_{n>=0} Product_{k=2*n..3*n-1} ((1+x)^k - B(x)) ;
A338183: 1 = Sum_{n>=0} Product_{k=3*n..4*n-1} ((1+x)^k - C(x)).
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} Product_{k=n..2*n-1} ( (1+x)^k - A(x) ).
(2) 1 = Sum_{n>=0} (1+x)^(n*(3*n-1)/2) / Product_{k=n..2*n} (1 + (1+x)^k*A(x)).
a(n) ~ c * d^n * n! / sqrt(n), where d = 4.6634423082484267335372948079179... and c = 0.10798828318774... - Vaclav Kotesovec, Oct 24 2020
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 23*x^3 + 383*x^4 + 8456*x^5 + 228657*x^6 + 7268077*x^7 + 264627570*x^8 + 10842464809*x^9 + 493454895161*x^10 + ...
Let A = A(x), then g.f. A(x) satisfies
1 = 1 + ((1+x) - A) + ((1+x)^2 - A)*((1+x)^3 - A) + ((1+x)^3 - A)*((1+x)^4 - A)*((1+x)^5 - A) + ((1+x)^4 - A)*((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A) + ((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A)*((1+x)^8 - A)*((1+x)^9 - A) + ... + Product_{k=n..2*n-1} ((1+x)^k - A(x)) + ...
also
1 = 1/(1 + A) + (1+x)/((1 + (1+x)*A)*(1 + (1+x)^2*A)) + (1+x)^5/((1 + (1+x)^2*A)*(1 + (1+x)^3*A)*(1 + (1+x)^4*A)) + (1+x)^12/((1 + (1+x)^3*A)*(1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)) + (1+x)^22/((1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)) + ... + (1+x)^(n*(3*n-1)/2)/( Product_{k=n..2*n} (1 + (1+x)^(k*A(x)) ) + ...
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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, prod(k=m, 2*m-1, (1+x)^k - Ser(A)) ) )[#A] ); A[n+1]}
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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