OFFSET
0,2
COMMENTS
Hankel transform is A176333.
Let A(n) denote the n X n array such that the i-th row of this array is the sequence obtained by applying the partial sum operator i-1 times to the tuple ((sqrt(-1))^m, 1 <= m <= n). Then the negative of the real part of the (n, n)-entry of A(n) equals a(n-2) for all n > 2. - John M. Campbell, Jan 20 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = Sum_{k=0..n} Sum_{j=0..n} binomial(j,n-k)*binomial(j,k)*(-1)^(n-j).
Logarithm g.f.: arctan(x*C(x)) = Sum_{n>=1} a(n)/n*x^n, where C(x) = (1-sqrt(1-4*x))/(2*x) (A000108). - Vladimir Kruchinin, Aug 10 2010
Conjecture: 6*n*a(n) + 2*(-17*n+10)*a(n-1) + (47*n-60)*a(n-2) + 10*(-3*n+5)*a(n-3) + 4*(2*n-5)*a(n-4) = 0. - R. J. Mathar, Nov 24 2012
Recurrence: 2*n*(5*n-8)*a(n) = 2*(25*n^2 - 50*n + 18)*a(n-1) - (45*n^2 - 92*n + 36)*a(n-2) + 2*(2*n-3)*(5*n-3)*a(n-3). - Vaclav Kotesovec, Feb 12 2014
a(n) ~ 4^(n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2 * c(x)^2) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^2) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024
a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n+k+2,n-k). - Seiichi Manyama, Nov 14 2025
MAPLE
a:=n->add(add(binomial(j, n-k)*binomial(j, k)*(-1)^(n-j), j=0..n), k=0..n): seq(a(n), n=0..30); # Muniru A Asiru, Jan 23 2019
MATHEMATICA
f[n_]:= (-1)^n*Sum[Binomial[n+k, k] Cos[Pi(n+k)/2], {k, 0, n}]; Array[f, 24, 0] (* Robert G. Wilson v, Apr 02 2012 *)
PROG
(PARI) {a(n) = sum(k=0, n, sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)* binomial(j, k))) }; vector(30, n, n--; a(n)) \\ G. C. Greubel, Feb 21 2019
(PARI) a(n) = {my(v = vector(n, k, I^k)); for (k=1, n-1, v = vector(n, i, sum(j=1, i, v[j])); ); -real(v[n]); } \\ Michel Marcus, Feb 25 2019
(PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k, n)); \\ Seiichi Manyama, Jan 29 2023
(PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^2*(2/(1+sqrt(1-4*x)))^2))) \\ Seiichi Manyama, Jan 29 2023
(Magma) [(&+[ (&+[(-1)^(n-j)*Binomial(j, n-k)*Binomial(j, k): j in [0..n]]): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 21 2019
(SageMath) [sum(sum((-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 21 2019
(GAP) List([0..30], n -> Sum([0..n], k -> Sum([0..n], j -> (-1)^(n-j)* Binomial(j, n-k)*Binomial(j, k) ))); # G. C. Greubel, Feb 22 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 15 2010
STATUS
approved