A326002 G.f.: Sum_{n>=0} (n+1) * x^n * (1 + x^n)^n.

1, 2, 5, 4, 11, 6, 22, 8, 29, 22, 41, 12, 89, 14, 71, 76, 109, 18, 214, 20, 196, 190, 155, 24, 573, 56, 209, 388, 519, 30, 877, 32, 809, 694, 341, 316, 2119, 38, 419, 1132, 2411, 42, 2045, 44, 2531, 2986, 599, 48, 6053, 106, 3011, 2500, 4759, 54, 4978, 4016, 6589, 3478, 929, 60, 21468, 62, 1055, 5524, 10713, 10076, 12046, 68, 13499, 6142, 18656, 72, 34474, 74, 1481, 29716, 20939, 5622, 28432, 80, 57921, 10000, 1805, 84, 84155, 42926, 1979, 12268, 41449, 90, 122339, 24116, 44759, 14974, 2351, 77616, 153969

Offset: 0

Author

Paul D. Hanna, Jun 01 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 some fixed integer k; here, k = 2 and p = 1, q = x, r = x.

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 4*x^3 + 11*x^4 + 6*x^5 + 22*x^6 + 8*x^7 + 29*x^8 + 22*x^9 + 41*x^10 + 12*x^11 + 89*x^12 + 14*x^13 + 71*x^14 + 76*x^15 + 109*x^16 + 18*x^17 + 214*x^18 + 20*x^19 + 196*x^20 + ...
where we have the following series identity:
A(x) = 1 + 2*x*(1+x) + 3*x^2*(1+x^2)^2 + 4*x^3*(1+x^3)^3 + 5*x^4*(1+x^4)^4 + 6*x^5*(1+x^5)^5  + 7*x^6*(1+x^6)^6 + 8*x^7*(1+x^7)^7 + 9*x^8*(1+x^8)^8 + 10*x^9*(1+x^9)^9 + ...
is equal to
A(x) = 1/(1-x)^2 + 2*x^2/(1-x^2)^3 + 3*x^6/(1-x^3)^4 + 4*x^12/(1-x^4)^5 + 5*x^20/(1-x^5)^6 + 6*x^30/(1-x^6)^7 + 7*x^42/(1-x^7)^8 + 8*x^56/(1-x^8)^9 + ...
		

Links

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

Crossrefs

Cf. A217668 (k=1), A326003 (k=3), A326004 (k=4), A326005 (k=5).

Programs

Maple

N:= 100: # for a(0)..a(N)
S:= series(add((n+1)*x^n*(1+x^n)^n,n=0..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Jun 03 2019

PARI

{a(n) = my(A = sum(m=0,n, (m+1) * x^m * (1 + x^m +x*O(x^n))^m)); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

PARI

{a(n) = my(A = sum(m=0,n, (m+1) * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+2))); polcoeff(A,n)}
for(n=0,120,print1(a(n),", "))

Formula

Generating functions.

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

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