A124376 - OEIS

%I #19 Feb 21 2025 05:39:28

%S 1,1,1,1,4,1,1,7,7,1,1,10,19,10,1,1,13,37,37,13,1,1,16,61,92,61,16,1,

%T 1,19,91,185,185,91,19,1,1,22,127,326,440,326,127,22,1,1,25,169,525,

%U 896,896,525,169,25,1,1,28,217,792,1638,2072,1638,792,217,28,1

%N Number triangle with column k generated by x^k*(1+2*k*x+C(k,2)*x^2)/(1-x)^(k+1).

%H Paolo Xausa, <a href="/A124376/b124376.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).

%F T(n,k) = Sum_{j=0..n} C(k,k-j)*C(n-j,k)*C(2,j)*[k<=n].

%F T(n,k) = T(n,n-k).

%e Triangle begins

%e   1,

%e   1,  1,

%e   1,  4,  1,

%e   1,  7,  7,   1,

%e   1, 10, 19,  10,   1,

%e   1, 13, 37,  37,  13,  1,

%e   1, 16, 61,  92,  61, 16,  1,

%e   1, 19, 91, 185, 185, 91, 19, 1

%t A124376[n_, k_] := Sum[Binomial[k, k-j]*Binomial[n-j, k]*Binomial[2, j], {j, 0, n}];

%t Table[A124376[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Feb 21 2025 *)

%o (PARI) C(i,j) =binomial(i,j);

%o T(n,k) = if (k<=n, sum(j=0, n, C(k,k-j)*C(n-j,k)*C(2,j)));

%o row(n) = vector(n+1, k, T(n,k-1));

%o for (n=0, 10, print(row(n))) \\ _Michel Marcus_, Feb 19 2025

%Y Columns include A016777, A003215, A096000.

%Y Cf. A158920.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Oct 28 2006

%E More terms from _Michel Marcus_, Feb 19 2025