A340358 - OEIS

%I #5 Jan 10 2021 11:50:52

%S 1,5,24,152,1094,8508,69565,588469,5106516,45199827,406485567,

%T 3703483221,34111556603,317103532465,2971283282979,28033510000286,

%U 266092385194061,2539244496436404,24346664830510834,234435203932318053,2266062742515203697,21980115620177318458

%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x))^3.

%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 = 2, and t = 3.

%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))^3.

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

%e G.f.: A(x) = 1 + 5*x + 24*x^2 + 152*x^3 + 1094*x^4 + 8508*x^5 + 69565*x^6 + 588469*x^7 + 5106516*x^8 + 45199827*x^9 + 406485567*x^10 + ...

%e where

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

%e also

%e A(x) = 1/(1 - x)^2 + 3*x*A(x)/(1 - x^2)^2 + 6*x^2*A(x)^2/(1 - x^3)^2 + 10*x^3*A(x)^3/(1 - x^4)^2 + 15*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))^3 )); 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, (m+1)*(m+2)/2 * 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, A340357, A340359, A340360.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 07 2021