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 some fixed integer k; here, k = 2 and p = 1, q = x, r = x.
LINKS
Robert Israel, Table of n, a(n) for n = 0..10000
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).
EXAMPLE
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 + ...
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
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Paul D. Hanna, Jun 01 2019
STATUS
approved