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A101164
Triangle read by rows: Delannoy numbers minus binomial coefficients.
8
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
0,8
LINKS
Eric Weisstein's World of Mathematics, Delannoy Number
Eric Weisstein's World of Mathematics, Binomial Coefficient
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)
EXAMPLE
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;
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 *)
PROG
(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
(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
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Dec 03 2004
STATUS
approved