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 A302565 G.f. A(x) satisfies: A(x) = Sum_n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1). 4
 1, 1, 7, 85, 1429, 30517, 792007, 24293389, 862902745, 34918162057, 1587910815271, 80217252865861, 4457823231346717, 270261899977497325, 17749585402744292215, 1255201826997862952845, 95083758340337074058545, 7680863233559647281837265, 659040900304099125516970375, 59855299015030039092312638965 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA G.f. A(x) satisfies: (1) A(x) = Sum_n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1). (2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A008548(n)*x^n, the o.g.f. of the quintuple factorials. (3) A(x) = 1 + x*A(x)^2 * (A(x) + 6*x*A'(x)) / (A(x) + x*A'(x)). (4) A(x) = 1/(1 - x*A(x)/(1 - 5*x*A(x)/(1 - 6*x*A(x)/(1 - 10*x*A(x)/(1 - 11*x*A(x)/(1 - 15*x*A(x)/(1 - 16*x*A(x)/(1 - ...)))))))), a continued fraction. a(n) ~ sqrt(2*Pi) * 5^n * n^(n - 3/10) / (Gamma(1/5) * exp(n - 1/5)). - Vaclav Kotesovec, Jun 18 2019 EXAMPLE G.f.: A(x) = 1 + x + 7*x^2 + 85*x^3 + 1429*x^4 + 30517*x^5 + 792007*x^6 + 24293389*x^7 + 862902745*x^8 + 34918162057*x^9 + ... such that A(x) = 1 + x*A(x) + 6*x^2*A(x)^2 + 66*x^3*A(x)^3 + 1056*x^4*A(x)^4 + 22176*x^5*A(x)^5 + ... + x^n*A(x)^n * Product_{k=0..n-1} (5*k + 1) + ... PROG (PARI) /* Series Reversion of Quintuple Factorials g.f.: */ {a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0, m-1, 5*k + 1)) +x^2*O(x^n)), n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* Differential Equation: */ {a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 6*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) /* Continued fraction: */ {a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(5*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A008548, A088368, A301363, A302100, A302535. Sequence in context: A064089 A216698 A317353 * A049412 A056547 A293055 Adjacent sequences:  A302562 A302563 A302564 * A302566 A302567 A302568 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 09 2018 STATUS approved

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Last modified August 13 05:41 EDT 2020. Contains 336442 sequences. (Running on oeis4.)