A101164 Triangle read by rows: Delannoy numbers minus binomial coefficients.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 7, 3, 0, 0, 4, 15, 15, 4, 0, 0, 5, 26, 43, 26, 5, 0, 0, 6, 40, 94, 94, 40, 6, 0, 0, 7, 57, 175, 251, 175, 57, 7, 0, 0, 8, 77, 293, 555, 555, 293, 77, 8, 0, 0, 9, 100, 455, 1079, 1431, 1079, 455, 100, 9, 0, 0, 10, 126, 668, 1911, 3191, 3191, 1911, 668, 126, 10, 0

Offset: 0

Author

Reinhard Zumkeller, Dec 03 2004

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  0
  0, 0;
  0, 1,  0;
  0, 2,  2,   0;
  0, 3,  7,   3,   0;
  0, 4, 15,  15,   4,   0;
  0, 5, 26,  43,  26,   5,  0;
  0, 6, 40,  94,  94,  40,  6, 0;
  0, 7, 57, 175, 251, 175, 57, 7, 0;
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of table, flattened
• Eric Weisstein's World of Mathematics, Delannoy Number
• Eric Weisstein's World of Mathematics, Binomial Coefficient
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A001477, A005449, A008288, A094706, A101165, A101166, A225413.

Programs

Haskell

a101164 n k = a101164_tabl !! n !! k
a101164_row n = a101164_tabl !! n
a101164_tabl = zipWith (zipWith (-)) a008288_tabl a007318_tabl
-- Reinhard Zumkeller, Jul 30 2013

Magma

A101164:= func< n,k | (&+[Binomial(n-k,j)*Binomial(k,j)*2^j: j in [0..n-k]]) - Binomial(n,k) >;
[A101164(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 17 2021

Mathematica

T[n_, k_]:= Hypergeometric2F1[-k, k-n, 1, 2] - Binomial[n, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2021 *)

Sage

def T(n,k): return simplify(hypergeometric([-n+k, -k], [1], 2)) - binomial(n,k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 17 2021

Formula

T(n, k) = A008288(n, k) - binomial(n, k), 0<=k<=n, where binomial=A007318.

T(n,2) = A005449(n-2) for n>1;

T(n,3) = A101165(n-3) for n>2;

T(n,4) = A101166(n-4) for n>3;

Sum_{k=0..n} T(n, k) = A094706(n).

From G. C. Greubel, Sep 17 2021: (Start)

T(n, k) = Sum_{j=0..n-k} binomial(n-k, j)*binomial(k, j)*2^j - binomial(n,k).

T(n, 1) = n-1, n > 0. (End)