A325998 G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (x + x^n)^n.

1, 6, 6, 22, 21, 51, 28, 126, 45, 170, 156, 246, 91, 627, 120, 496, 588, 876, 190, 1626, 231, 1776, 1536, 1392, 325, 4977, 798, 2086, 3405, 5025, 496, 8694, 561, 8122, 6636, 4086, 3881, 21597, 780, 5440, 11781, 26016, 946, 24114, 1035, 28001, 33348, 8976, 1225, 70302, 2586, 36946, 30501, 56127, 1540, 66318, 46698, 82056, 45660, 16710, 1891, 268242, 2016, 20032, 79806, 140106, 122398, 171738, 2415, 180835, 92256, 249612, 2701, 482532, 2850, 32566

Offset

0,2

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,

(2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - p*q^n*r)^(n+k),

for any fixed integer k; here, k = 3 and q = x, p = x, r = 1.

Links

Table of n, a(n) for n=0..75.

Formula

G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * (x + x^n)^n.

G.f.: Sum_{n>=0} (n+1)*(n+2)/2 * x^(n^2) / (1 - x^(n+1))^(n+3).

Example

G.f.: A(x) = 1 + 6*x + 6*x^2 + 22*x^3 + 21*x^4 + 51*x^5 + 28*x^6 + 126*x^7 + 45*x^8 + 170*x^9 + 156*x^10 + 246*x^11 + 91*x^12 + 627*x^13 + 120*x^14 +...
where
A(x) = 1 + 3*(x + x) + 6*(x + x^2)^2 + 10*(x + x^3)^3 + 15*(x + x^4)^4 + 21*(x + x^5)^5 + 28*(x + x^6)^6 + 36*(x + x^7)^7 + 45*(x + x^8)^8 + 55*(x + x^9)^9 + ...
Also
A(x) = 1/(1-x)^3 + 3*x/(1 - x^2)^4 + 6*x^4/(1 - x^3)^5 + 10*x^9/(1 - x^4)^6 + 15*x^16/(1 - x^5)^7 + 21*x^25/(1 - x^6)^8 + 28*x^36/(1 - x^7)^9 + 36*x^49/(1 - x^8)^10 + 45*x^64/(1 - x^9)^11 + 55*x^81/(1 - x^10)^12 + ...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2 * (x + x^m +x*O(x^n))^m), n)}
for(n=0, 100, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2 * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3)), n)}
for(n=0, 100, print1(a(n), ", "))

Crossrefs

Cf. A217669, A325997, A325999.

Sequence in context: A253069 A255460 A255464 * A322216 A053168 A141388

Adjacent sequences: A325995 A325996 A325997 * A325999 A326000 A326001

Keyword

nonn

Author

Paul D. Hanna, Jun 02 2019

Status

approved