A326004 G.f.: Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * (1 + x^n)^n.

1, 4, 14, 20, 55, 56, 154, 120, 305, 280, 566, 364, 1189, 560, 1520, 1376, 2429, 1140, 4570, 1540, 5226, 4544, 6304, 2600, 14685, 3556, 10934, 11980, 18215, 4960, 31882, 5984, 31289, 27160, 27150, 12636, 82093, 9880, 39920, 55160, 93631, 13244, 121178, 15180, 126875, 130696, 78224, 19600, 316645, 22940, 165386, 179844, 281399, 27720, 370090, 150976, 410629, 297560, 179830, 37820, 1208458, 41664, 229184, 489280, 801305, 450516, 987482, 54740

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 = 4 and p = 1, q = x, r = x.

Links

Table of n, a(n) for n=0..67.

Formula

Generating functions.

(1) Sum_{n>=0} (n+1)*(n+2)*(n+3)/3! * x^n * (1 + x^n)^n.

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

Example

G.f.: A(x) = 1 + 4*x + 14*x^2 + 20*x^3 + 55*x^4 + 56*x^5 + 154*x^6 + 120*x^7 + 305*x^8 + 280*x^9 + 566*x^10 + 364*x^11 + 1189*x^12 + 560*x^13 + 1520*x^14 + 1376*x^15 + 2429*x^16 + 1140*x^17 + 4570*x^18 + 1540*x^19 + 5226*x^20 + ...
where we have the following series identity:
A(x) = 1 + 4*x*(1+x) + 10*x^2*(1+x^2)^2 + 20*x^3*(1+x^3)^3 + 35*x^4*(1+x^4)^4 + 56*x^5*(1+x^5)^5  + 84*x^6*(1+x^6)^6 + 120*x^7*(1+x^7)^7 + 165*x^8*(1+x^8)^8 + 220*x^9*(1+x^9)^9 +...
is equal to
A(x) = 1/(1-x)^4 + 4*x^2/(1-x^2)^5 + 10*x^6/(1-x^3)^6 + 20*x^12/(1-x^4)^7 + 35*x^20/(1-x^5)^8 + 56*x^30/(1-x^6)^9 + 84*x^42/(1-x^7)^10 + 120*x^56/(1-x^8)^11 +...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, (m+1)*(m+2)*(m+3)/3! * 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)/3! * x^m * x^(m^2) / (1 - x^(m+1) +x*O(x^n))^(m+4))); polcoeff(A, n)}
for(n=0, 120, print1(a(n), ", "))

Crossrefs

Cf. A217668 (k=1), A326002 (k=2), A326003 (k=3), A326005 (k=5).

Sequence in context: A051448 A075319 A030470 * A297358 A267768 A185008

Adjacent sequences: A326001 A326002 A326003 * A326005 A326006 A326007

Keyword

nonn

Author

Paul D. Hanna, Jun 01 2019

Status

approved