%I #35 Nov 09 2022 19:23:05
%S 1,7,52,403,3206,25954,212738,1760035,14666470,122920642,1035046816,
%T 8749594462,74207078908,631140253072,5381022869822,45975731083555,
%U 393556869530630,3374504760608026,28977403637496104,249167023897718138
%N a(n) = Sum_{i=0..n} (2^(i)*(-1)^(i+n)*C(n,i)*C(2*n+i-1,n-1)).
%H G. C. Greubel, <a href="/A259554/b259554.txt">Table of n, a(n) for n = 1..1000</a>
%H V. V. Kruchinin and D. V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T_{2n,n} in Triangles</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
%F G.f.: A(x) = x*B(x)'/B(x), where B(x) is g.f. of A003168.
%F Recurrence: 4*n*(2*n-1)*(17*n^2 - 51*n + 38)*a(n) = (1207*n^4 - 4828*n^3 + 6659*n^2 - 3662*n + 672)*a(n-1) - 2*(n-2)*(2*n-3)*(17*n^2 - 17*n + 4)*a(n-2). - _Vaclav Kotesovec_, Jul 01 2015
%F a(n) ~ (71+17*sqrt(17))^n / (17^(1/4) * sqrt(Pi*n) * 2^(4*n+1)). - _Vaclav Kotesovec_, Jul 01 2015
%F a(n) = (1/2)*Sum_{k = 0..n} binomial(n-1,n-k)*binomial(2*n+k-1,k). - _Vladimir Kruchinin_, Oct 07 2016
%F a(n) = n*hypergeom([2*n+1, -n+1], [2], -1). - _Peter Luschny_, Oct 07 2016
%F From _Peter Bala_, Nov 08 2022: (Start)
%F a(n) = (1/2)*[x^n] ( (1 - x)/(1 - 2*x) )^(2*n). Cf. A002002(n) = [x^n] ( (1 - x)/(1 - 2*x) )^n.
%F a(n) = (1/2)*Sum_{k = 0..n} (-1)^(n-k)*2^k*binomial(2*n,n-k)*binomial(2*n+k-1,k).
%F a(n) = (1/2)*(-1)^n*binomial(2*n,n)*hypergeom( [-n, 2*n], [n+1], 2).
%F The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p >= 3 and all positive integers n and r. (End)
%p a := n -> n*hypergeom([2*n+1, -n+1], [2], -1):
%p seq(simplify(a(n)), n=1..9); # _Peter Luschny_, Oct 07 2016
%t Table[Sum[2^i * (-1)^(i+n) * Binomial[n, i] * Binomial[2*n+i-1, n-1], {i, 0, n}], {n,1,20}] (* _Vaclav Kotesovec_, Jul 01 2015 *)
%o (Maxima) a(n):=sum(2^(i)*(-1)^(i+n)*binomial(n,i)*binomial(2*n+i-1,n-1),i,0,n);
%o (PARI) a(n) = sum(i=0, n, 2^i*(-1)^(i+n)*binomial(n, i)*binomial(2*n+i-1, n-1)); \\ _Michel Marcus_, Jul 02 2015
%Y Cf. A002002, A003168.
%K nonn,easy
%O 1,2
%A _Vladimir Kruchinin_, Jun 30 2015