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A340357 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)). 4

%I

%S 1,3,7,20,64,232,908,3717,15716,67996,299396,1337022,6040421,27556567,

%T 126762966,587324586,2738338960,12837950292,60483207417,286206067039,

%U 1359678614745,6482510515788,31006901328525,148750651958227,715545729962692,3450638733403489

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)).

%C The g.f. A(x) of this sequence is motivated by the following identity:

%C Sum_{n>=0} C(t+n-1,n) * p^n/(1 - q*r^n)^s = Sum_{n>=0} C(s+n-1,n) * q^n/(1 - p*r^n)^t ;

%C here, p = x, q = x*A(x), r = x, s = 1, and t = 2.

%F G.f. A(x) satisfies the following relations.

%F (1) A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)).

%F (2) A(x) = Sum_{n>=0} x^n * A(x)^n / (1 - x^(n+1))^2.

%e G.f.: A(x) = 1 + 3*x + 7*x^2 + 20*x^3 + 64*x^4 + 232*x^5 + 908*x^6 + 3717*x^7 + 15716*x^8 + 67996*x^9 + 299396*x^10 + ...

%e where

%e A(x) = 1/(1 - x*A(x)) + 2*x/(1 - x^2*A(x)) + 3*x^2/(1 - x^3*A(x)) + 4*x^3/(1 - x^4*A(x)) + 5*x^4/(1 - x^5*A(x)) + ...

%e also

%e A(x) = 1/(1 - x)^2 + x*A(x)/(1 - x^2)^2 + x^2*A(x)^2/(1 - x^3)^2 + x^3*A(x)^3/(1 - x^4)^2 + x^4*A(x)^4/(1 - x^5)^2 + ...

%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (m+1) * x^m / (1 - x^(m+1)*A +x*O(x^n)) )); polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m * A^m / (1 - x^(m+1) +x*O(x^n))^2 )); ; polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A340329, A340338, A340355, A340356, A340358, A340359, A340360.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 07 2021

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Last modified January 31 17:49 EST 2023. Contains 359980 sequences. (Running on oeis4.)