A118192 Antidiagonal sums of triangle A118190: a(n) = Sum_{k=0..floor(n/2)} 5^(k*(n-2*k)) for n>=0.

1, 1, 2, 6, 27, 151, 1252, 18876, 421877, 11797501, 489062502, 36867190626, 4119892578127, 576049853531251, 119400024902343752, 45003894807128984376, 25145828723919677734377, 17579646409034759521875001

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/(1-5^n*x^2).

a(2*n) = Sum_{k=0..n} 5^(2*k*(n-k)).

a(2*n+1) = Sum_{k=0..n} 5^(k*(2*(n-k)+1)).

Example

A(x) = 1/(1-x^2) + x/(1-5*x^2) + x^2/(1-25*x^2) + x^3/(1-125*x^2) + ...
  = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 151*x^5 + ...

Mathematica

Table[Sum[5^(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, (5^k)^(n-2*k) )
(Magma) [(&+[5^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021
(Sage) [sum(5^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Crossrefs

Cf. A118190 (triangle), A118191 (row sums).

Sequence in context: A030932 A340356 A363456 * A338180 A307595 A058133

Adjacent sequences: A118189 A118190 A118191 * A118193 A118194 A118195

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2006

Status

approved