A337721 G.f.: Sum_{n>=0} (1 + x*(1+x)^n)^n * x^n.

1, 1, 2, 4, 9, 23, 62, 179, 549, 1773, 6003, 21233, 78187, 298894, 1183387, 4842221, 20438964, 88849325, 397183838, 1823456223, 8587051052, 41434641992, 204654311282, 1033757421996, 5335693879201, 28118977852767, 151192761513229, 828884087889407

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Formula

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

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

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 23*x^5 + 62*x^6 + 179*x^7 + 549*x^8 + 1773*x^9 + 6003*x^10 + 21233*x^11 + 78187*x^12 + ...
where
A(x) = 1 + (1 + x*(1+x))*x + (1 + x*(1+x)^2)^2*x^2 + (1 + x*(1+x)^3)^3*x^3 + (1 + x*(1+x)^4)^4*x^4 + ... + (1 + x*(1+x)^n)^n*x^n + ...
also
A(x) = 1/(1 - x) + (1+x)*x^2/(1 - x*(1+x))^2 + (1+x)^4*x^4/(1 - x*(1+x)^2)^3 + (1+x)^9*x^6/(1 - x*(1+x)^3)^4 + (1+x)^16*x^8/(1 - x*(1+x)^4)^5 + (1+x)^25*x^10/(1 - x*(1+x)^5)^6  + ... + (1+x)^(n^2)*x^(2*n)/(1 - x*(1+x)^n)^(n+1) + ...

Mathematica

nmax = 30; CoefficientList[Series[Sum[(1 + x*(1+x)^k)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 17 2020 *)

Prog

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

Crossrefs

Cf. A337720.

Sequence in context: A190277 A274882 A127384 * A369328 A058585 A001573

Adjacent sequences: A337718 A337719 A337720 * A337722 A337723 A337724

Keyword

nonn

Author

Paul D. Hanna, Sep 17 2020

Status

approved