A326286 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.

1, 8, 41, 90, 671, 6788, 31803, 119486, 746315, 1959108, 17687917, 168219722, 1612302467, 6734874480, 30113355681, 146636111898, 714115126295, 4578149141156, 16402101919131, 158506042034472, 1074010290985493, 7994020873236474, 64888090981118585, 366989246419220666, 1682317245914363391, 6686668206846701272, 28987038620286638765, 149983846501792016730, 1140728507133902950163, 7842482827496240439354, 30507352871067667404773

Offset

0,2

Comments

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

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

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

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

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).

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).

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)).

Example

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.,
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 + ...
and writing G(x) as a power series in x starting as
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 + ...
then the coefficients of x^(n*(n+1)) in G(x) form this sequence.

Prog

(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)}
a(n) = A326285(n*(n+1))
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A326285, A323679.

Sequence in context: A093104 A120646 A041116 * A250322 A135797 A171714

Adjacent sequences: A326283 A326284 A326285 * A326287 A326288 A326289

Keyword

nonn

Author

Paul D. Hanna, Jul 01 2019

Status

approved