A325997 G.f.: Sum_{n>=0} (n+1) * (x + x^n)^n.

1, 4, 3, 10, 8, 18, 7, 40, 9, 44, 41, 54, 13, 150, 15, 88, 127, 168, 19, 298, 21, 324, 275, 180, 25, 854, 132, 238, 524, 774, 31, 1286, 33, 1180, 893, 378, 674, 2998, 39, 460, 1406, 3744, 43, 2790, 45, 3458, 4397, 648, 49, 8420, 303, 4714, 2960, 6270, 55, 7060, 6492, 9120, 4049, 990, 61, 30748, 63, 1120, 7697, 13788, 15082, 17626, 69, 16834, 6971, 28788, 73, 48088, 75, 1558, 39792, 25578, 12091, 41578, 81, 77874, 11540, 1890, 85, 121650, 58227

Offset: 0

Author

Paul D. Hanna, Jun 02 2019

Keywords

• nonn
• look

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 = 2 and p = x, q = x, r = 1.

Examples

			G.f.: A(x) = 1 + 4*x + 3*x^2 + 10*x^3 + 8*x^4 + 18*x^5 + 7*x^6 + 40*x^7 + 9*x^8 + 44*x^9 + 41*x^10 + 54*x^11 + 13*x^12 + 150*x^13 + 15*x^14 + 88*x^15 +...
where
A(x) = 1 + 2*(x + x) + 3*(x + x^2)^2 + 4*(x + x^3)^3 + 5*(x + x^4)^4 + 6*(x + x^5)^5 + 7*(x + x^6)^6 + 8*(x + x^7)^7 + 9*(x + x^8)^8 + 10*(x + x^9)^9 + ...
Also
A(x) = 1/(1-x)^2 + 2*x/(1 - x^2)^3 + 3*x^4/(1 - x^3)^4 + 4*x^9/(1 - x^4)^5 + 5*x^16/(1 - x^5)^6 + 6*x^25/(1 - x^6)^7 + 7*x^36/(1 - x^7)^8 + 8*x^49/(1 - x^8)^9 + 9*x^64/(1 - x^9)^10 + 10*x^81/(1 - x^10)^11 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A217669, A325998, A325999.

Programs

Maple

G:= add((n+1)*(x+x^n)^n,n=0..100):
S:= series(G,x,101):
seq(coeff(S,x,n),n=0..100); # Robert Israel, Jun 02 2019

PARI

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

Formula

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

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