A325999 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * (x + x^n)^n.

1, 8, 10, 40, 45, 116, 84, 320, 165, 520, 496, 868, 455, 2100, 680, 2136, 2264, 3680, 1330, 6920, 1771, 7988, 6920, 8060, 2925, 22732, 4914, 13580, 17365, 26440, 5456, 46212, 6545, 45000, 37800, 32376, 20773, 119660, 10660, 46900, 74221, 143528, 14190, 161540, 16215, 177196, 194764, 89800, 20825, 447040, 28046, 239928, 229725, 384860, 29260, 492128, 257734, 569140, 372480, 201500, 39711, 1763416, 43680, 255200, 639430, 1068856, 733074, 1337080

Offset: 0

Author

Paul D. Hanna, Jun 02 2019

Keywords

• nonn

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 = 4 and q = x, p = x, r = 1.

Examples

			G.f.: A(x) = 1 + 8*x + 10*x^2 + 40*x^3 + 45*x^4 + 116*x^5 + 84*x^6 + 320*x^7 + 165*x^8 + 520*x^9 + 496*x^10 + 868*x^11 + 455*x^12 + 2100*x^13 + 680*x^14 +...
where
A(x) = 1 + 4*(x + x) + 10*(x + x^2)^2 + 20*(x + x^3)^3 + 35*(x + x^4)^4 + 56*(x + x^5)^5 + 84*(x + x^6)^6 + 120*(x + x^7)^7 + 165*(x + x^8)^8 + 220*(x + x^9)^9 + ...
Also
A(x) = 1/(1-x)^4 + 4*x/(1 - x^2)^5 + 10*x^4/(1 - x^3)^6 + 20*x^9/(1 - x^4)^7 + 35*x^16/(1 - x^5)^8 + 56*x^25/(1 - x^6)^9 + 84*x^36/(1 - x^7)^10 + 120*x^49/(1 - x^8)^11 + 165*x^64/(1 - x^9)^12 + 220*x^81/(1 - x^10)^13 + ...
		

Crossrefs

Cf. A217669, A325997, A325998.

Programs

PARI

{a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)*(m+3)/3! * (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)*(m+3)/3! * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+4)), n)}
for(n=0, 100, print1(a(n), ", "))

Formula

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

G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^(n^2) / (1 - x^(n+1))^(n+4).

FORMULAS FOR TERMS.

a(5*n + 2) = 0 (mod 5),

a(5*n + 3) = 0 (mod 5),

a(5*n + 4) = 0 (mod 5), for n >= 0.