A326005 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)*(n+4)/4! * x^n * (1 + x^n)^n.

1, 5, 20, 35, 100, 126, 330, 330, 775, 820, 1631, 1365, 3535, 2380, 5370, 5136, 9085, 5985, 16900, 8855, 21966, 19580, 29965, 17550, 60375, 24381, 58345, 57205, 90350, 40920, 152837, 52360, 164145, 141120, 175560, 93801, 404500, 101270, 280175, 309050, 503041, 148995, 714435, 178365, 748705, 708946, 633950, 249900, 1771645, 295135, 1120236, 1155015, 1760500, 395010, 2483110, 905576, 2622545, 2036060, 1744525, 595665, 6962328, 677040, 2343880

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.

Links

Table of n, a(n) for n=0..62.

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

Cf. A217668 (k=1), A326002 (k=2), A326003 (k=3), A326004 (k=4).

Sequence in context: A003339 A047716 A344190 * A063110 A044066 A013337

Adjacent sequences: A326002 A326003 A326004 * A326006 A326007 A326008

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2019

Status

approved