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 = 5 and p = 1, q = x, r = x.
FORMULA
Generating functions.
(1) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n.
(2) Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^(n*(n+1)) / (1 - x^(n+1))^(n+5).
EXAMPLE
G.f.: A(x) = 1 + 5*x + 20*x^2 + 35*x^3 + 100*x^4 + 126*x^5 + 330*x^6 + 330*x^7 + 775*x^8 + 820*x^9 + 1631*x^10 + 1365*x^11 + 3535*x^12 + 2380*x^13 + 5370*x^14 + 5136*x^15 + 9085*x^16 + 5985*x^17 + 16900*x^18 + 8855*x^19 + 21966*x^20 + ...
where we have the following series identity:
A(x) = 1 + 5*x*(1+x) + 15*x^2*(1+x^2)^2 + 35*x^3*(1+x^3)^3 + 70*x^4*(1+x^4)^4 + 126*x^5*(1+x^5)^5 + 210*x^6*(1+x^6)^6 + 330*x^7*(1+x^7)^7 + 495*x^8*(1+x^8)^8 + 715*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^5 + 5*x^2/(1-x^2)^6 + 15*x^6/(1-x^3)^7 + 35*x^12/(1-x^4)^8 + 70*x^20/(1-x^5)^9 + 126*x^30/(1-x^6)^10 + 210*x^42/(1-x^7)^11 + 330*x^56/(1-x^8)^12 +...
PROG
(PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)*(m+3)*(m+4)/4! * 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)*(m+3)*(m+4)/4! * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+5))); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 01 2019
STATUS
approved