A118191 Row sums of triangle A118190: a(n) = Sum_{k=0..n} 5^(k*(n-k)) for n>=0.

1, 2, 7, 52, 877, 32502, 2740627, 507843752, 214111484377, 198376465625002, 418186492923828127, 1937270172119160156252, 20419262349796295263671877, 472966350615029335022460937502, 24925857360591180741786959228515627

Offset: 0

Author

Paul D. Hanna, Apr 15 2006

Keywords

• nonn

Comments

Self-convolution of A118195; in general, sqrt(Sum_{n>=0} x^n/(1-q^n*x)) is an integer series whenever q == 1 (mod 4). Also equals column 0 of the matrix square of triangle A118190, where [A118190^2](n,k) = a(n-k)*5^(k*(n-k)) for n>=k>=0.

Examples

			A(x) = 1/(1-x) + x/(1-5*x) + x^2/(1-25*x) + x^3/(1-125*x) + ...
  = 1 + 2*x + 7*x^2 + 52*x^3 + 877*x^4 + 32502*x^5 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..70

Crossrefs

Cf. A118190 (triangle), A118192 (antidiagonal sums), A118195 (A(x)^(1/2)).

Programs

Magma

[(&+[5^(k*(n-k)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021

Mathematica

Table[Sum[5^(k*(n-k)), {k,0,n}], {n,0,30}] (* G. C. Greubel, Jun 29 2021 *)

PARI

a(n)=sum(k=0, n, (5^k)^(n-k))

Sage

[sum(5^(k*(n-k)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Formula

G.f.: A(x) = Sum_{n>=0} x^n/(1-5^n*x).

a(n) ~ c * 5^(n^2/4), where c = EllipticTheta[3, 0, 1/5] (in Mathematica) = JacobiTheta3(0,1/5) (in Maple) = 1.40320102401310720671088653743895... if n is even and c = EllipticTheta[2, 0, 1/5] = JacobiTheta2(0,1/5) = 1.39106543858832939481476315485543... if n is odd. - Vaclav Kotesovec, Aug 20 2025