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A124376 Number triangle with column k generated by x^k*(1+2*k*x+C(k,2)*x^2)/(1-x)^(k+1).

%I A124376 #19 Feb 21 2025 05:39:28
%S A124376 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 A124376 1,19,91,185,185,91,19,1,1,22,127,326,440,326,127,22,1,1,25,169,525,
%U A124376 896,896,525,169,25,1,1,28,217,792,1638,2072,1638,792,217,28,1
%N A124376 Number triangle with column k generated by x^k*(1+2*k*x+C(k,2)*x^2)/(1-x)^(k+1).
%H A124376 Paolo Xausa, <a href="/A124376/b124376.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%F A124376 T(n,k) = Sum_{j=0..n} C(k,k-j)*C(n-j,k)*C(2,j)*[k<=n].
%F A124376 T(n,k) = T(n,n-k).
%e A124376 Triangle begins
%e A124376   1,
%e A124376   1,  1,
%e A124376   1,  4,  1,
%e A124376   1,  7,  7,   1,
%e A124376   1, 10, 19,  10,   1,
%e A124376   1, 13, 37,  37,  13,  1,
%e A124376   1, 16, 61,  92,  61, 16,  1,
%e A124376   1, 19, 91, 185, 185, 91, 19, 1
%t A124376 A124376[n_, k_] := Sum[Binomial[k, k-j]*Binomial[n-j, k]*Binomial[2, j], {j, 0, n}];
%t A124376 Table[A124376[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Feb 21 2025 *)
%o A124376 (PARI) C(i,j) =binomial(i,j);
%o A124376 T(n,k) = if (k<=n, sum(j=0, n, C(k,k-j)*C(n-j,k)*C(2,j)));
%o A124376 row(n) = vector(n+1, k, T(n,k-1));
%o A124376 for (n=0, 10, print(row(n))) \\ _Michel Marcus_, Feb 19 2025
%Y A124376 Columns include A016777, A003215, A096000.
%Y A124376 Cf. A158920.
%K A124376 easy,nonn,tabl
%O A124376 0,5
%A A124376 _Paul Barry_, Oct 28 2006
%E A124376 More terms from _Michel Marcus_, Feb 19 2025