A341375 - OEIS

%I #10 Feb 15 2021 22:41:34

%S 1,4,22,56,181,256,730,968,1859,2368,4810,4680,9674,9944,15024,17580,

%T 29693,25792,47226,42476,62505,66388,104586,83700,141472,132768,

%U 181046,173964,273121,202124,359578,307860,410700,399924,551552,436980,743931,629304,816002

%N G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^4 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^4.

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^4 = Sum_{n>=0} a(n) * x^n / (1 - x^(n+1))^4.

%F (2) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^4 = Sum_{n>=0} (n+1)*(n+2)*(n+3)/6 * x^n * A( x^(n+1) ).

%e A(x) = 1 + 4*x + 22*x^2 + 56*x^3 + 181*x^4 + 256*x^5 + 730*x^6 + 968*x^7 + 1859*x^8 + 2368*x^9 + 4810*x^10 + 4680*x^11 + 9674*x^12 + ...

%e such that

%e D(x)^4 = 1/(1-x)^4 + 4*x/(1-x^2)^4 + 22*x^2/(1-x^3)^4 + 56*x^3/(1-x^4)^4 + 181*x^4/(1-x^5)^4 + 256*x^5/(1-x^6)^4 + ... + a(n)*x^n/(1-x^(n+1))^4 + ...

%e and

%e D(x)^4 = A(x) + 4*x*A(x^2) + 10*x^2*A(x^3) + 20*x^3*A(x^4) + 35*x^4*A(x^5) + 56*x^5*A(x^6) + 84*x^6*A(x^7) + ... + (n+1)*(n+2)*(n+3)/6*x^n*A(x^(n+1)) + ...

%e where

%e D(x)^4 = 1 + 8*x + 32*x^2 + 92*x^3 + 216*x^4 + 440*x^5 + 814*x^6 + 1392*x^7 + 2244*x^8 + 3452*x^9 + 5096*x^10 + 7292*x^11 + 10129*x^12 + ...

%e D(x) = 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 4*x^9 + 2*x^10 + 6*x^11 + 2*x^12 + ... + A000005(n+1)*x^n + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = polcoeff( sum(n=0,#A, x^n/(1 - x^(n+1) +x*O(x^#A)) )^4 - sum(n=0,#A-1,A[n+1]*x^n/(1 - x^(n+1) + x*O(x^#A))^4 ), #A-1) );A[n+1]}

%o for(n=0,100,print1(a(n),", "))

%Y Cf. A341373, A341374, A000005.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 11 2021