A357157 - OEIS

%I #9 Sep 23 2022 03:11:21

%S 1,1,1,1,7,1,1,1,22,1,1,-19,57,1,1,1,22,1,1,1,303,-349,1,1,463,1,-593,

%T 1,793,1,1,-2204,2584,1,1,1,-2287,1,3082,1,3004,-8084,1,1,14465,-3674,

%U -14299,1,6189,1,22276,-24023,-2056,1,1,1,18714,1,1,-34985,24305,-60059,87517,1,20350

%N Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(4*n) * (1 - x^n)^(n-2).

%C Related identities:

%C (I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).

%C (I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).

%C (I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).

%C (I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).

%C (I.5) 0 = Sum_{n=-oo..+oo} binomial(n+k-1, k) * x^(k*n) * (1 - x^n)^(n-1) for fixed positive integer k.

%C (I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).

%C (I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).

%C (I.8) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)/3! * x^(n*(n+2)) / (1 - x^(n+2))^(n+3).

%C (I.9) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)*(n+3)/4! * x^(n*(n+3)) / (1 - x^(n+3))^(n+4).

%C (I.10) 0 = Sum_{n=-oo..+oo} (-1)^n * binomial(n+k-1, k) * x^(n*(n+k-1)) / (1 - x^(n+k-1))^(n+k) for fixed positive integer k.

%C (I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)*(n+2)*(n+3)/120 * x^(4*n) * (1 - x^n)^(n-2).

%C (I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n+1)*n*(n-1)*(n-2)*(n-3)/120 * x^(n*(n-2)) / (1 - x^n)^(n+2).

%H Paul D. Hanna, <a href="/A357157/b357157.txt">Table of n, a(n) for n = 4..2050</a>

%F G.f. A(x) = Sum_{n>=4} a(n)*x^n satisfies:

%F (1) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(4*n) * (1 - x^n)^(n-2).

%F (2) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(5*n) * (1 - x^n)^(n-2).

%F (3) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)*(n+4)/120 * x^(4*n) * (1 - x^n)^(n-2).

%F (4) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-3)) / (1 - x^n)^(n+2).

%F (5) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-2)) / (1 - x^n)^(n+2).

%F (6) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)*(n-3)*(n-4)/120 * x^(n*(n-2)) / (1 - x^n)^(n+2).

%e G.f.: A(x) = x^4 + x^5 + x^6 + x^7 + 7*x^8 + x^9 + x^10 + x^11 + 22*x^12 + x^13 + x^14 - 19*x^15 + 57*x^16 + x^17 + x^18 + x^19 + 22*x^20 + ...

%e where

%e A(x) = ... + 5*x^(-20)*(1 - x^(-5))^(-7) + 1*x^(-16)*(1 - x^(-4))^(-6) + 0*x^(-12) + 0*x^(-8) + 0*x^(-4) + 0 + 1*x^4/(1-x) + 5*x^8 + 15*x^12*(1 - x^3) + 35*x^16*(1 - x^4)^2 + 70*x^20*(1 - x^5)^3 + ... + n*(n+1)*(n+2)*(n+3)/24 * x^(4*n)*(1 - x^n)^(n-2) + ...

%o (PARI) {a(n) = my(A = sum(m=-n-1,n+1, if(m==0,0, m*(m+1)*(m+2)*(m+3)/24 * x^(4*m) * (1 - x^m +x*O(x^n))^(m-2) )) );

%o polcoeff(A,n)}

%o for(n=4,100,print1(a(n),", "))

%Y Cf. A291937, A356774, A356775, A357156.

%K sign

%O 4,5

%A _Paul D. Hanna_, Sep 22 2022