A118187 Antidiagonal sums of triangle A118185: a(n) = Sum_{k=0..[n/2]} 4^(k*(n-2*k)) for n>=0.

1, 1, 2, 5, 18, 81, 514, 5185, 73730, 1327361, 33685506, 1359217665, 77311508482, 5567355555841, 565149010231298, 91215553426898945, 20753150033413537794, 5977902509385249259521, 2427296516310194305630210

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..100

Formula

G.f.: A(x) = Sum_{n>=0} x^n/(1-4^n*x^2).

a(2*n) = Sum_{k=0..n} 4^(2*k*(n-k)).

a(2*n+1) = Sum_{k=0..n} (4^k)^(2*(n-k)+1).

Example

A(x) = 1/(1-x^2) + x/(1-4*x^2) + x^2/(1-16*x^2) + x^3/(1-64*x^2) + ...
  = 1 + x + 2*x^2 + 5*x^3 + 18*x^4 + 81*x^5 + 514*x^6 + ...

Mathematica

Table[Sum[4^(k*(n-2*k)), {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)

Prog

(PARI) a(n)=sum(k=0, n\2, (4^k)^(n-2*k) )
(Magma) [(&+[4^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021
(Sage) [sum(4^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Crossrefs

Cf. A118185 (triangle), A118186 (row sums).

Sequence in context: A111916 A328440 A308634 * A307773 A332776 A038720

Adjacent sequences: A118184 A118185 A118186 * A118188 A118189 A118190

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2006

Status

approved