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

0,2

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.

Links

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

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

Example

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)

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A076863 A191742 A181082 * A317080 A075272 A353536

Adjacent sequences: A118183 A118184 A118185 * A118187 A118188 A118189

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2006

Status

approved