A338793 G.f.: Sum_{n>=0} ( Product_{k=1..n} 1 + x*(1+x)^k ) / 2^(n+1).

1, 1, 3, 9, 35, 157, 799, 4525, 28131, 189961, 1381731, 10752725, 89024543, 780441893, 7215286619, 70101824033, 713576518099, 7589578829637, 84143768397175, 970341357523749, 11616886603985203, 144132396583436329, 1850345648687946571, 24543468645365759797

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

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

Example

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 35*x^4 + 157*x^5 + 799*x^6 + 4525*x^7 + 28131*x^8 + 189961*x^9 + 1381731*x^10 + 10752725*x^11 + 89024543*x^12 + ...
where
A(x) = 1/2 + (1 + x*(1+x))/2^2 + (1 + x*(1+x))*(1 + x*(1+x)^2)/2^3 + (1 + x*(1+x))*(1 + x*(1+x)^2)*(1 + x*(1+x)^3)/2^4 + (1 + x*(1+x))*(1 + x*(1+x)^2)*(1 + x*(1+x)^3)*(1 + x*(1+x)^4)/2^5 + ... + (Product_{k=1..n} 1 + x*(1+x)^k)/2^(n+1) + ...
Also,
A(x) = 1 + x*(1+x)/(2 - (1+x)) + x^2*(1+x)^3/((2 - (1+x))*(2 - (1+x)^2)) + x^3*(1+x)^6/((2 - (1+x))*(2 - (1+x)^2)*(2 - (1+x)^3)) + x^4*(1+x)^10/((2 - (1+x))*(2 - (1+x)^2)*(2 - (1+x)^3)*(2 - (1+x)^4)) + x^5*(1+x)^15/((2 - (1+x))*(2 - (1+x)^2)*(2 - (1+x)^3)*(2 - (1+x)^4)*(2 - (1+x)^5)) + ... + x^n*(1+x)^(n*(n+1)/2)/(Product_{k=0..n} 2 - (1+x)^k) + ...

Prog

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

Crossrefs

Sequence in context: A030268 A097277 A034428 * A370341 A101880 A222398

Adjacent sequences: A338790 A338791 A338792 * A338794 A338795 A338796

Keyword

nonn

Author

Paul D. Hanna, Nov 09 2020

Status

approved