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 A352235 G.f. A(x) satisfies: A(x) = 1 + x*A(x) / (A(x) - 3*x*A'(x)). 4
 1, 1, 3, 24, 309, 5262, 108894, 2618718, 71246145, 2154788970, 71563126710, 2586270267600, 100995812044266, 4237522832234832, 190126298040192912, 9085093650185205498, 460711407231295513689, 24715373661154672634058, 1398648334415007990887454 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..378 FORMULA G.f. A(x) satisfies: (1) [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2) for n >= 1. (2) A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)). (3) A'(x) = A(x) * (1 + x/(1 - A(x))) / (3*x). (4) A(x) = exp( Integral (1 + x/(1 - A(x))) / (3*x) dx ). a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.09232038797888963484135336... - Vaclav Kotesovec, Nov 16 2023 EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 309*x^4 + 5262*x^5 + 108894*x^6 + 2618718*x^7 + 71246145*x^8 + ... such that A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)). Related table. The table of coefficients of x^k in A(x)^(3*n+2) begins: n=0: [1, 2, 7, 54, 675, 11286, 230742, ...]; n=1: [1, 5, 25, 190, 2210, 34981, 688635, ...]; n=2: [1, 8, 52, 416, 4642, 69872, 1322848, ...]; n=3: [1, 11, 88, 759, 8349, 120549, 2195886, ...]; n=4: [1, 14, 133, 1246, 13790, 193060, 3391017, ...]; n=5: [1, 17, 187, 1904, 21505, 295154, 5017618, ...]; n=6: [1, 20, 250, 2760, 32115, 436524, 7217250, ...]; ... in which the following pattern holds: [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2), n >= 1, as illustrated by [x^1] A(x)^2 = 2 = [x^0] 2*A(x)^2 = 2*1; [x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5; [x^3] A(x)^8 = 416 = [x^2] 8*A(x)^8 = 8*52; [x^4] A(x)^11 = 8349 = [x^3] 11*A(x)^11 = 11*759; [x^5] A(x)^14 = 193060 = [x^4] 14*A(x)^14 = 14*13790; [x^6] A(x)^17 = 5017618 = [x^5] 17*A(x)^17 = 17*295154; ... PROG (PARI) /* Using A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)) */ {a(n) = my(A=1); for(i=1, n, A = 1 + x*A/(A - 3*x*A' + x*O(x^n)) ); polcoeff(A, n)} for(n=0, 20, print1(a(n), ", ")) (PARI) /* Using [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2)*A(x)^(3*n+2) */ {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff((x*Ser(A)^(3*(#A-2)+2) - Ser(A)^(3*(#A-2)+2)/(3*(#A-2)+2)), #A-1)); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A088715, A286797, A317352, A352236, A352237, A352238. Sequence in context: A258301 A365584 A138209 * A075142 A138428 A047056 Adjacent sequences: A352232 A352233 A352234 * A352236 A352237 A352238 KEYWORD nonn AUTHOR Paul D. Hanna, Mar 08 2022 STATUS approved

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Last modified April 19 16:52 EDT 2024. Contains 371794 sequences. (Running on oeis4.)