A188202 - OEIS

%I #14 Mar 07 2017 00:14:05

%S 1,2,18,560,68614,34210752,69223161876,564393502852608,

%T 18462508476328312902,2418515748615522678533120,

%U 1267759405004680431879193624828,2658525712652053771685500828954042368

%N Central coefficients in (1 + 2^n*x + x^2)^n.

%C Variant of the central trinomial coefficients: A002426(n) = [x^n] (1+x+x^2)^n.

%H G. C. Greubel, <a href="/A188202/b188202.txt">Table of n, a(n) for n = 0..57</a>

%F a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(n-k, n-2k) * 2^(n*(n-2k)).

%F Equals the logarithmic derivative of A188203 (ignoring initial term).

%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Feb 12 2015

%e Ignoring the initial term, this sequence forms the logarithmic series:

%e L(x) = 2*x + 18*x^2/2 + 560*x^3/3 + 68614*x^4/4 + 34210752*x^5/5 + ...

%e where the g.f. of A188203 begins:

%e exp(L(x)) = 1 + 2*x + 11*x^2 + 206*x^3 + 17586*x^4 + 6878604*x^5 + ...

%e Illustrate definition.

%e The coefficients of x^k in (1 + 2^n*x + x^2)^n, k=0..2n, n>=0, begin:

%e n=0: [(1)];

%e n=1: [1, (2), 1];

%e n=2: [1, 8, (18), 8, 1];

%e n=3: [1, 24, 195, (560), 195, 24, 1];

%e n=4: [1, 64, 1540, 16576, (68614), 16576, 1540, 64, 1];

%e n=5: [1, 160, 10245, 328320, 5273610, (34210752), 5273610, 328320, ...];

%e n=6: [1, 384, 61446, 5244800, 251904015, 6458183424, (69223161876), ...];

%e n=7: [1, 896, 344071, 73405696, 9396961301, 721848120448, 30814514741283, (564393502852608), ...]; ...

%e where the above central coefficients in parenthesis form this sequence.

%t Table[Sum[Binomial[n,k] * Binomial[n-k, n-2k] * 2^(n*(n-2k)), {k,0,Floor[n/2]}], {n,0,20}] (* _Vaclav Kotesovec_, Feb 11 2015 *)

%o (PARI) {a(n)=polcoeff((1+2^n*x+x^2)^n,n)}

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

%Y Cf. A188203 (exp); variants: A002426, A186925.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 24 2011