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%I #25 Jul 08 2019 04:51:36
%S 1,3,9,15,79,657,2789,9679,50187,122379,911783,7942511,71320919,
%T 292307479,1254424307,5649367163,25471489371,151196109585,
%U 513167692755,4381764642655,29243835423341,209688880294187,1635709191538759,9050289283302795,40510463137400579,159199144373303265,672166886239228581,3264163837045382109,22816461252418921287,152055680315368539533,570344377866479916661,3765597736734681752239,26013973013767849104195
%N a(n) equals the coefficient of x^(n*(n+1)) in Sum_{m>=0} x^m * (1 + x^m)^m / (1 + x^(m+1))^(m+1) for n >= 0.
%C a(n) = A323557(n*(n+1)) for n >= 0.
%C Appears to consist of all the odd terms in A323557.
%H Paul D. Hanna, <a href="/A323679/b323679.txt">Table of n, a(n) for n = 0..180</a>
%F a(n) = [x^(n*(n+1))] Sum_{k>=0} x^k * (1 + x^k)^k / (1 + x^(k+1))^(k+1).
%F a(n) = [x^(n*(n+1))] Sum_{k>=0} (-x)^k * (1 - x^k)^k / (1 - x^(k+1))^(k+1).
%F a(n) = [x^(n*(n+1))] Sum_{m>=0} x^m * Sum_{k=0..m} binomial(m,k) * (x^m - x^k)^(m-k).
%F a(n) = [x^(n*(n+1))] Sum_{m>=0} x^m * Sum_{k=0..m} binomial(m,k) * (-1)^k * (x^m + x^k)^(m-k).
%F a(n) = [x^(n*(n+1))] Sum_{m>=0} x^m * Sum_{k=0..m} binomial(m,k) * (-1)^k * Sum_{j=0..m-k} binomial(m-k,j) * x^((m-k)*(m-j)).
%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 + ... + A323557(n)*x^n + ...
%e then the odd coefficients of x^n in G(x), occurring at n = k*(k+1) for k>=0, form this sequence.
%e RELATED SEQUENCE.
%e The terms A323557(n*(n+3)) begin
%e [1, 2, 12, 50, 72, 142, 5346, 38338, 240902, 1369462, ..., A323678(n), ...].
%t nmax = 20; s = CoefficientList[Series[Sum[x^m*(1 + x^m)^m/(1 + x^(m + 1))^(m + 1), {m, 0, nmax*(nmax + 1)}], {x, 0, nmax*(nmax + 1)}], x]; Table[s[[n*(n + 1) + 1]], {n, 0, nmax}] (* _Vaclav Kotesovec_, Jul 08 2019 *)
%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+1))
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A323557, A323678, A323677.
%Y Cf. A326286.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 03 2019
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