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Coefficients of x^(n*(n+3)) in Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1).
4

%I #17 Feb 06 2019 21:58:32

%S 1,2,12,50,72,142,5346,38338,240902,1369462,8927272,29594702,78001922,

%T 259042422,2690290778,26069217364,144738683318,959061370318,

%U 7256969189340,35982546596434,223064363341712,1127137948649166,4200987237905836,14295780646046930,50369139372748128,286016211643945372,2979062704489818940,30112757810084509370,153633113062169831324,927474622673328916402

%N Coefficients of x^(n*(n+3)) in Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1).

%C a(n) = A323557(n*(n+3)) for n >= 0.

%H Paul D. Hanna, <a href="/A323678/b323678.txt">Table of n, a(n) for n = 0..179</a>

%F a(n) = [x^(n*(n+3))] Sum_{k>=0} x^k * (1 + x^k)^k / (1 + x^(k+1))^(k+1).

%F a(n) = [x^(n*(n+3))] Sum_{k>=0} (-x)^k * (1 - x^k)^k / (1 - x^(k+1))^(k+1).

%e Given the g.f. of A323557, G(x) = Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1), i.e.,

%e G(x) = 1/(1 + x) + x*(1 + x)/(1 + x^2)^2 + x^2*(1 + x^2)^2/(1 + x^3)^3 + x^3*(1 + x^3)^3/(1 + x^4)^4 + x^4*(1 + x^4)^4/(1 + x^5)^5 + x^5*(1 + x^5)^5/(1 + x^6)^6 + x^6*(1 + x^6)^6/(1 + x^7)^7 + x^7*(1 + x^7)^7/(1 + x^8)^8 + ...

%e and writing G(x) as a power series in x starting as

%e G(x) = 1 + 3*x^2 - 2*x^3 + 2*x^4 + 9*x^6 - 14*x^7 + 8*x^8 + 12*x^10 - 12*x^11 + 15*x^12 - 52*x^13 + 76*x^14 - 36*x^15 + 2*x^16 + 50*x^18 - 104*x^19 + 79*x^20 - 140*x^21 + 324*x^22 - 276*x^23 + 128*x^24 - 144*x^25 + 118*x^26 - 28*x^27 + 72*x^28 - 336*x^29 + 657*x^30 - 802*x^31 + 1184*x^32 - 1568*x^33 + 1086*x^34 - 288*x^35 + 302*x^36 - 1032*x^37 + 1212*x^38 - 480*x^39 + 142*x^40 + ... + A323557(n)*x^n + ...

%e then the coefficients of x^(n*(n+3)) in G(x), for n >= 0, form this sequence.

%e RELATED SEQUENCE.

%e The terms A323557(n*(n+1)) begin

%e [1, 3, 9, 15, 79, 657, 2789, 9679, 50187, 122379, 911783, ..., A323679(n), ...]

%e and appear to consist of only odd numbers.

%o (PARI) {A323557(n) = my(A=sum(m=0, n, x^m * (1 + x^m +x*O(x^n))^m/(1 + x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}

%o a(n) = A323557(n*(n+3))

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

%Y Cf. A323557, A323679, A323677.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 03 2019