%I #8 Jul 01 2019 23:38:32

%S 1,8,41,90,671,6788,31803,119486,746315,1959108,17687917,168219722,

%T 1612302467,6734874480,30113355681,146636111898,714115126295,

%U 4578149141156,16402101919131,158506042034472,1074010290985493,7994020873236474,64888090981118585,366989246419220666,1682317245914363391,6686668206846701272,28987038620286638765,149983846501792016730,1140728507133902950163,7842482827496240439354,30507352871067667404773

%N a(n) equals the coefficient of x^(n*(n+1)) in Sum_{m>=0} (m+1) * x^m * (1 + x^m)^m / (1 + x^(m+1))^(m+2) for n >= 0.

%C a(n) = A326285(n*(n+1)) for n >= 0.

%C a(2*n) = 1 (mod 2) for n > 0.

%C It appears that all the odd terms in A326285 occur at even positions in this sequence.

%H Paul D. Hanna, <a href="/A326286/b326286.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = [x^(n*(n+1))] Sum_{k>=0} (k+1) * x^k * (1 + x^k)^k / (1 + x^(k+1))^(k+2).

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

%F a(n) = [x^(n*(n+1))] Sum_{m>=0} (m+1) * 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} (m+1) * 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} (m+1) * 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 A326285, G(x) = Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+2), i.e.,

%e G(x) = 1/(1 + x)^2 + 2*x*(1 + x)/(1 + x^2)^3 + 3*x^2*(1 + x^2)^2/(1 + x^3)^4 + 4*x^3*(1 + x^3)^3/(1 + x^4)^5 + 5*x^4*(1 + x^4)^4/(1 + x^5)^6 + 6*x^5*(1 + x^5)^5/(1 + x^6)^7 + 7*x^6*(1 + x^6)^6/(1 + x^7)^8 + 8*x^7*(1 + x^7)^7/(1 + x^8)^9 + ...

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

%e G(x) = 1 + 8*x^2 - 6*x^3 + 10*x^4 + 41*x^6 - 64*x^7 + 48*x^8 + 82*x^10 - 84*x^11 + 90*x^12 - 300*x^13 + 532*x^14 - 284*x^15 + 34*x^16 + 428*x^18 - 892*x^19 + 671*x^20 - 960*x^21 + 2620*x^22 - 2440*x^23 + 1184*x^24 - 1440*x^25 + 1408*x^26 - 420*x^27 + 618*x^28 - 3024*x^29 + 6788*x^30 - 8274*x^31 + 11022*x^32 + ...

%e then the coefficients of x^(n*(n+1)) in G(x) form this sequence.

%o (PARI) {A326285(n) = my(A=sum(m=0, n, (m+1) * x^m * (1 + x^m +x*O(x^n))^m/(1 + x^(m+1) +x*O(x^n))^(m+2) )); polcoeff(A, n)}

%o a(n) = A326285(n*(n+1))

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

%Y Cf. A326285, A323679.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 01 2019