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 any fixed integer k; here, k = 3 and p = 1, q = x, r = x.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..2050
FORMULA
Generating functions.
(1) Sum_{n>=0} (n+1)*(n+2)/2 * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1)*(n+2)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+3).
EXAMPLE
G.f.: A(x) = 1 + 3*x + 9*x^2 + 10*x^3 + 27*x^4 + 21*x^5 + 64*x^6 + 36*x^7 + 105*x^8 + 85*x^9 + 171*x^10 + 78*x^11 + 359*x^12 + 105*x^13 + 372*x^14 + 346*x^15 + 573*x^16 + 171*x^17 + 1105*x^18 + 210*x^19 + 1116*x^20 + ...
where we have the following series identity:
A(x) = 1 + 3*x*(1+x) + 6*x^2*(1+x^2)^2 + 10*x^3*(1+x^3)^3 + 15*x^4*(1+x^4)^4 + 21*x^5*(1+x^5)^5 + 28*x^6*(1+x^6)^6 + 36*x^7*(1+x^7)^7 + 45*x^8*(1+x^8)^8 + 55*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^3 + 3*x^2/(1-x^2)^4 + 6*x^6/(1-x^3)^5 + 10*x^12/(1-x^4)^6 + 15*x^20/(1-x^5)^7 + 21*x^30/(1-x^6)^8 + 28*x^42/(1-x^7)^9 + 36*x^56/(1-x^8)^10 +...
PROG
(PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)/2 * 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)*(m+2)/2 * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+3))); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 01 2019
STATUS
approved