A253273 Triangle T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1), read by rows.

1, 1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 12, 14, 5, 1, 6, 18, 30, 25, 6, 1, 7, 25, 53, 66, 41, 7, 1, 8, 33, 84, 136, 132, 63, 8, 1, 9, 42, 124, 244, 315, 245, 92, 9, 1, 10, 52, 174, 400, 636, 673, 428, 129, 10, 1, 11, 63, 235, 615, 1152, 1522, 1346, 711, 175, 11

Offset: 0

Author

Vladimir Kruchinin, May 01 2015

Keywords

• nonn
• tabl

Examples

			The triangle begins as:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  7,   4;
  1,  5, 12,  14,   5;
  1,  6, 18,  30,  25,   6;
  1,  7, 25,  53,  66,  41,   7;
  1,  8, 33,  84, 136, 132,  63,   8;
  1,  9, 42, 124, 244, 315, 245,  92,   9;
  1, 10, 52, 174, 400, 636, 673, 428, 129, 10;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A095263.

Programs

Magma

T:= func< n,k | (&+[Binomial(k+j,k-j+1)*Binomial(n-k,j-1): j in [0..n-k+1]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 17 2021

Mathematica

T[n_, k_]:= Sum[Binomial[k+j,k-j+1]*Binomial[n-k,j-1], {j,0,n-k+1}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 17 2021 *)

Maxima

T(n,m):=sum(binomial(m+k,m-k+1)*binomial(n-m,k-1),k,0,n-m+1);

Sage

def T(n,k): return sum(binomial(k+j,k-j+1)*binomial(n-k,j-1) for j in (0..n-k+1))
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 17 2021

Formula

T(n,k) = Sum_{j=0..n-k+1} binomial(k+j,k-j+1)*binomial(n-k,j-1).

Sum_{k=0..n} T(n,k) = A095263(n+1).

G.f.: 1/( (1-x)*(1+y^2) - (2-x)*y ).