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 A338178 G.f.: A(x) satisfies 1 = Sum_{n>=0} Product_{k=n..2*n-1} ( (1+x)^k - A(x) ). 3
 1, 1, 2, 23, 383, 8456, 228657, 7268077, 264627570, 10842464809, 493454895161, 24695915073150, 1348165575801780, 79740451015249487, 5080946666286668723, 347046897108675507548, 25300042941361404342404, 1960975025091063811051791, 161045636717926154800016147, 13970534533583992498224467231 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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)). LINKS Paul D. Hanna, Table of n, a(n) for n = 0..200 FORMULA 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 EXAMPLE 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)) ) + ... PROG (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), ", ")) CROSSREFS Cf. A338181, A338183. Sequence in context: A239109 A266923 A060941 * A219890 A119774 A074649 Adjacent sequences: A338175 A338176 A338177 * A338179 A338180 A338181 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 14 2020 STATUS approved

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Last modified December 8 14:17 EST 2022. Contains 358693 sequences. (Running on oeis4.)