A326000 G.f.: Sum_{n>=0} (n+1) * ((1+x)^n - 1)^n.

1, 2, 12, 120, 1607, 26862, 536816, 12466468, 329648274, 9774030812, 321057111308, 11570735358300, 453874209520951, 19248243764760562, 877497573254643438, 42791783608096161848, 2222646606788322292656, 122500263059540271947448, 7140154262067048381368062, 438819217371889984410077532, 28360033818941846664929891481, 1922734355204851243123303962324

Offset: 0

Author

Paul D. Hanna, Jun 01 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 = 2 and p = -1, q = 1+x, r = 1.

Examples

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 120*x^3 + 1607*x^4 + 26862*x^5 + 536816*x^6 + 12466468*x^7 + 329648274*x^8 + 9774030812*x^9 + 321057111308*x^10 + ...
such that
A(x) = 1 + 2*((1+x)-1) + 3*((1+x)^2-1)^2 + 4*((1+x)^3-1)^3 + 5*((1+x)^4-1)^4 + 6*((1+x)^5-1)^5  + 7*((1+x)^6-1)^6 + 8*((1+x)^7-1)^7 + 9*((1+x)^8-1)^8 + 10*((1+x)^9-1)^9 +...
is equal to
A(x) = 1/2^2 + 2*(1+x)/(1+(1+x))^3 + 3*(1+x)^4/(1+(1+x)^2)^4 + 4*(1+x)^9/(1+(1+x)^3)^5 + 5*(1+x)^16/(1+(1+x)^4)^6 + 6*(1+x)^25/(1+(1+x)^5)^7 + 7*(1+x)^36/(1+(1+x)^6)^8 + 8*(1+x)^49/(1+(1+x)^7)^9 + ...
		

Crossrefs

Cf. A122400, A326001.

Programs

PARI

{a(n) = my(A = sum(m=0,n, (m+1) * ((1+x)^m - 1 +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

Generating functions.

(1) Sum_{n>=0} (n+1) * ((1+x)^n - 1)^n.

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