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A300989 O.g.f. A(x) satisfies: A(x) = x*(1 - 3*x*A'(x)) / (1 - 4*x*A'(x)). 7

%I

%S 1,1,6,50,520,6312,86080,1288704,20862720,361454720,6652338176,

%T 129341001216,2645494627328,56734280221696,1272300911597568,

%U 29769957834147840,725430667245355008,18379623419316338688,483476314203202945024,13187069277429966733312,372512001057014648537088,10886129458069912361631744,328776894530826384975593472

%N O.g.f. A(x) satisfies: A(x) = x*(1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).

%C O.g.f. equals the logarithm of the e.g.f. of A300988.

%C The e.g.f. G(x) of A300988 satisfies: [x^n] G(x)^(4*n) = (n+3) * [x^(n-1)] G(x)^(4*n) for n>=1.

%H Paul D. Hanna, <a href="/A300989/b300989.txt">Table of n, a(n) for n = 1..200</a>

%F O.g.f. A(x) satisfies: [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) for n>=1.

%F a(n) ~ c * n! * n^7, where c = 0.00000132855349... - _Vaclav Kotesovec_, Mar 20 2018

%e O.g.f.: A(x) = x + x^2 + 6*x^3 + 50*x^4 + 520*x^5 + 6312*x^6 + 86080*x^7 + 1288704*x^8 + 20862720*x^9 + 361454720*x^10 + ...

%e where

%e A(x) = x * (1 - 3*x*A'(x)) / (1 - 4*x*A'(x)).

%e RELATED SERIES.

%e exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1369*x^4/4! + 69561*x^5/5! + 4991371*x^6/6! + 471516403*x^7/7! + 56029153713*x^8/8! + 8112993527089*x^9/9! + ... + A300988(n)*x^n/n! + ...

%e A'(x) = 1 + 2*x + 18*x^2 + 200*x^3 + 2600*x^4 + 37872*x^5 + 602560*x^6 + 10309632*x^7 + 187764480*x^8 + 3614547200*x^9 + ...

%o (PARI) {a(n) = my(A=x); for(i=1, n, A = x*(1-3*x*A')/(1-4*x*A' +x*O(x^n))); polcoeff(A, n)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) /* [x^n] exp( 4*n * A(x) ) = (n + 3) * [x^(n-1)] exp( 4*n * A(x) ) */

%o {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^(4*(#A-1))); A[#A] = ((#A+2)*V[#A-1] - V[#A])/(4*(#A-1)) ); polcoeff( log(Ser(A)), n)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A300988, A088716, A300736, A300987, A300991, A300993.

%Y Cf. A300591, A296171, A300593, A300595.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Mar 17 2018

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Last modified April 15 04:04 EDT 2021. Contains 342974 sequences. (Running on oeis4.)