A338182 G.f.: Sum_{n>=0} x^n * Product_{k=3*n..4*n-1} (1 + (1+x)^k).

1, 2, 7, 37, 251, 1947, 17141, 167163, 1779029, 20456722, 251997747, 3303602265, 45842747207, 670308495256, 10288071997990, 165203260586951, 2767484369773904, 48243983405917665, 873228719542308898, 16378723735806496553, 317781161715431909527, 6367610715649281689454, 131581260677384045968633

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: Sum_{n>=0} x^n * Product_{k=3*n..4*n-1} (1 + (1+x)^k).

G.f.: Sum_{n>=0} x^n * (1+x)^(n*(7*n-1)/2) / ( Product_{k=3*n..4*n} 1 - x*(1+x)^k ).

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 37*x^3 + 251*x^4 + 1947*x^5 + 17141*x^6 + 167163*x^7 + 1779029*x^8 + 20456722*x^9 + 251997747*x^10 + ...
where
A(x) = 1 + x*(1 + (1+x)^3) + x^2*(1 + (1+x)^6)*(1 + (1+x)^7) + x^3*(1 + (1+x)^9)*(1 + (1+x)^10)*(1 + (1+x)^11) + x^4*(1 + (1+x)^12)*(1 + (1+x)^13)*(1 + (1+x)^14)*(1 + (1+x)^15) + x^5*(1 + (1+x)^15)*(1 + (1+x)^16)*(1 + (1+x)^17)*(1 + (1+x)^18)*(1 + (1+x)^19) + ... + x^n*Product_{k=3*n..4*n-1} (1 + (1+x)^k) + ...
Also
A(x) = 1/(1 - x)  +  x*(1+x)^3/((1 - x*(1+x)^3)*(1 - x*(1+x)^4))  +  x^2*(1+x)^13/((1 - x*(1+x)^6)*(1 - x*(1+x)^7)*(1 - x*(1+x)^8))  +  x^3*(1+x)^30/((1 - x*(1+x)^9)*(1 - x*(1+x)^10)*(1 - x*(1+x)^11)*(1 - x*(1+x)^12))  +  x^4*(1+x)^54/((1 - x*(1+x)^12)*(1 - x*(1+x)^13)*(1 - x*(1+x)^14)*(1 - x*(1+x)^15)*(1 - x*(1+x)^16)) + ... + x^n*(1+x)^(n*(7*n-1)/2)/(Product_{k=3*n..4*n} 1 - x*(1+x)^k) + ...

Prog

(PARI) {a(n) = polcoeff( sum(m=0, n, x^m * prod(k=3*m, 4*m-1, 1 + (1+x)^k +x*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = polcoeff( sum(m=0, n, x^m * (1+x +x*O(x^n))^(m*(7*m-1)/2) / prod(k=3*m, 4*m, 1 - x*(1+x)^k +x*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A338179, A338180.

Sequence in context: A125191 A300559 A302859 * A135164 A321087 A072597

Adjacent sequences: A338179 A338180 A338181 * A338183 A338184 A338185

Keyword

nonn

Author

Paul D. Hanna, Oct 19 2020

Status

approved