A117403 - OEIS

%I #15 Sep 08 2022 08:45:24

%S 1,1,2,3,6,13,34,105,386,1681,8706,53793,395266,3442753,35659778,

%T 440672385,6476038146,112812130561,2336999211010,57759810847233,

%U 1697654543745026,59146046307566593,2450521284684021762

%N a(n) = Sum_{k=0..floor(n/2)} 2^((n-2*k)*k) for n>=0.

%C Equals the antidiagonal sums of triangle A117401.

%H G. C. Greubel, <a href="/A117403/b117403.txt">Table of n, a(n) for n = 0..150</a>

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

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

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

%F G.f.: 1/(1-x^2) - x/(Q(0) +x-x^3), where Q(k) = x^2*(2+x)*2^k -1-x - x*(2*x^2*2^k -1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 11 2013

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 13*x^5 + 34*x^6 + 105*x^7 + ...

%e where

%e A(x) = 1/(1-x^2) + x/(1-2*x^2) + x^2/(1-4*x^2) + x^3/(1-8*x^2) + x^4/(1-16*x^2) + ...

%t Table[Sum[2^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Jun 28 2021 *)

%o (PARI) a(n) = sum(k=0,n\2,2^((n-2*k)*k))

%o (PARI) {a(n) = polcoeff(sum(m=0,n,x^m/(1-2^m*x^2 +x*O(x^n))),n)}

%o for(n=0,30,print1(a(n),", "))

%o (Magma) [(&+[2^(k*(n-2*k)) : k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jun 28 2021

%o (Sage) [sum(2^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # _G. C. Greubel_, Jun 28 2021

%Y Cf. A117401 (triangle), A117402 (row sums).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 12 2006

%E Name changed by _Paul D. Hanna_, Nov 11 2013