cp's OEIS Frontend

A306405 T(n,k) = k*Sum_{i=0..(n-k)/2} C(k,2*k+2*i-n)*C(k+2*i-1,i)/(k+i), triangle read by rows for n >= 1 and 1 <= k <= n.

%I A306405 #13 Dec 10 2023 18:09:52
%S A306405 1,1,1,1,2,1,1,3,3,1,2,4,6,4,1,2,7,10,10,5,1,5,10,18,20,15,6,1,5,19,
%T A306405 30,39,35,21,7,1,14,28,55,72,75,56,28,8,1,14,56,93,136,151,132,84,36,
%U A306405 9,1,42,84,174,248,300,288,217,120,45,10,1
%N A306405 T(n,k) = k*Sum_{i=0..(n-k)/2} C(k,2*k+2*i-n)*C(k+2*i-1,i)/(k+i), triangle read by rows for n >= 1 and 1 <= k <= n.
%F A306405 G.f.: 1/(1-y*(x*(1+x)*(1-sqrt(1-4*x^2))/(2*x^2)))-1.
%e A306405     1;
%e A306405     1,  1;
%e A306405     1,  2,  1;
%e A306405     1,  3,  3,  1;
%e A306405     2,  4,  6,  4,  1;
%e A306405     2,  7, 10, 10,  5, 1;
%e A306405     5, 10, 18, 20, 15, 6, 1;
%p A306405 # Seen as a (0,0)-based triangle:
%p A306405 gf := (2*(x + 1))/(sqrt(1 - 4*x^2) - 2*x*(x + 1)*y + 1):
%p A306405 serx := series(gf, x, 20): sery := n -> series(coeff(serx, x, n), y, 20):
%p A306405 row := n -> seq(coeff(sery(n), y, j), j=0..n):
%p A306405 seq(lprint(row(n)), n=0..9); # _Peter Luschny_, Feb 14 2019
%o A306405 (Maxima)
%o A306405 T(n,k):=k*sum((binomial(k,2*k+2*i-n)*binomial(k+2*i-1,i))/(k+i),i,0,(n-k)/2);
%Y A306405 Cf. A000108, A274115.
%K A306405 nonn,tabl
%O A306405 1,5
%A A306405 _Vladimir Kruchinin_, Feb 13 2019