A137638 Antidiagonal sums of square array A137634.
1, 3, 15, 72, 361, 1840, 9505, 49578, 260540, 1377328, 7316373, 39020372, 208809544, 1120621368, 6029023185, 32507001876, 175604614108, 950233307930, 5149691511432, 27946158749572, 151843410356906, 825949622559366
Offset: 0
Keywords
Programs
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PARI
{a(n)=sum(k=0,n,sum(j=0,k,binomial(2*j+n-k,j)*binomial(2*j+n-k,k-j)))} /* Using the g.f.: */ {a(n)=local(G=sqrt(1 - 4*x*(1+x)^2 +x*O(x^n))); polcoeff(2*(1+x)/((1+2*x+G)*G),n)}
Formula
G.f.: A(x) = 2*(1+x)/((1+2*x + G(x))*G(x)) where G(x) = sqrt(1 - 4*x*(1+x)^2).
a(n) = Sum_{k=0..n} Sum_{j=0..k} C(n-k+2*j,j)*C(n-k+2*j,k-j).
D-finite with recurrence 2*(n+1)*a(n) +(-3*n-7)*a(n-1) +2*(-17*n+10)*a(n-2) +8*(-7*n+10)*a(n-3) +2*(-18*n+37)*a(n-4) +4*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Jun 23 2023