A118181 Row sums of triangle A118180: a(n) = Sum_{k=0..n} (3^k)^(n-k) for n>=0.

1, 2, 5, 20, 137, 1622, 33293, 1182440, 72811793, 7757988842, 1433154521621, 458101483131260, 253879024041595289, 243453910296759945662, 404765167247068325944349, 1164432505878183620543030480

Offset

0,2

Comments

Also equals column 0 of the matrix square of triangle A118180, where [A118180^2](n,k) = a(n-k)*(3^k)^(n-k) for n>=k>=0.

Links

G. C. Greubel, Table of n, a(n) for n = 0..90

Formula

G.f.: A(x) = Sum_{n>=0} x^n/(1-3^n*x).

Example

A(x) = 1/(1-x) + x/(1-3x) + x^2/(1-9x) + x^3/(1-27x) + ...
= 1 + 2*x + 5*x^2 + 20*x^3 + 137*x^4 + 1622*x^5 + 33293*x^6 +...

Maple

seq( add(3^(k*(n-k)), k=0..n), n=0..30); # modified by G. C. Greubel, Jun 29 2021

Mathematica

Table[Sum[3^(k*(n-k)), {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)

Prog

(PARI) a(n)=sum(k=0, n, (3^k)^(n-k) );
(Magma) [(&+[3^(k*(n-k)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021
(Sage) [sum(3^(k*(n-k)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Crossrefs

Cf. A118180 (triangle), A118182 (antidiagonal sums); A118183, A118184.

Sequence in context: A012317 A297630 A297629 * A140988 A136650 A229662

Adjacent sequences: A118178 A118179 A118180 * A118182 A118183 A118184

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2006

Status

approved