A118186 Row sums of triangle A118185: a(n) = Sum_{k=0..n} 4^(k*(n-k)) for n>=0.

1, 2, 6, 34, 386, 8706, 395266, 35659778, 6476038146, 2336999211010, 1697654543745026, 2450521284684021762, 7120479243447937531906, 41112924905741324849774594, 477847273163370530909175414786

Offset: 0

Author

Paul D. Hanna, Apr 15 2006

Keywords

• nonn

Comments

Also equals column 0 of the matrix square of triangle A118185, where [A118185^2](n,k) = a(n-k)*4^(k*(n-k)) for n >= k >= 0.

Examples

			A(x) = 1/(1-x) + x/(1-4x) + x^2/(1-16x) + x^3/(1-64x) + ...
  = 1 + 2*x + 6*x^2 + 34*x^3 + 386*x^4 + 8706*x^5 + ...
From _Paul D. Hanna_, Oct 14 2009: (Start)
Another g.f.: (1 + x/2^1 + x^2/2^4 + x^3/2^9 + x^4/2^16 + ...)^2
  = 1 + 2*x/2^1 + 6*x^2/2^4 + 34*x^3/2^9 + 386*x^4/2^16 + ... (End)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..80

Crossrefs

Cf. A118185 (triangle), A118187 (antidiagonal sums).

Programs

Magma

[(&+[4^(k*(n-k)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021

Mathematica

Table[Sum[4^(k*(n-k)), {k,0,n}], {n,0,30}] (* G. C. Greubel, Jun 29 2021 *)

PARI

a(n)=sum(k=0, n, (4^k)^(n-k) );

PARI

{a(n)=2^(n^2)*polcoeff(sum(m=0,n,x^m/2^(m^2)+x*O(x^n))^2,n)} \\ Paul D. Hanna, Oct 14 2009

Sage

[sum(4^(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/(1-4^n*x).

G.f.: Sum_{n>=1} a(n)*x^n/2^(n^2) = ( Sum_{n>=0} x^n/2^(n^2) )^2. - Paul D. Hanna, Oct 14 2009