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

%I A101164 #36 Feb 16 2025 08:32:55
%S A101164 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,
%T A101164 94,94,40,6,0,0,7,57,175,251,175,57,7,0,0,8,77,293,555,555,293,77,8,0,
%U A101164 0,9,100,455,1079,1431,1079,455,100,9,0,0,10,126,668,1911,3191,3191,1911,668,126,10,0
%N A101164 Triangle read by rows: Delannoy numbers minus binomial coefficients.
%H A101164 Reinhard Zumkeller, <a href="/A101164/b101164.txt">Rows n = 0..100 of table, flattened</a>
%H A101164 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DelannoyNumber.html">Delannoy Number</a>
%H A101164 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialCoefficient.html">Binomial Coefficient</a>
%H A101164 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A101164 T(n, k) = A008288(n, k) - binomial(n, k), 0<=k<=n, where binomial=A007318.
%F A101164 T(n,2) = A005449(n-2) for n>1;
%F A101164 T(n,3) = A101165(n-3) for n>2;
%F A101164 T(n,4) = A101166(n-4) for n>3;
%F A101164 Sum_{k=0..n} T(n, k) = A094706(n).
%F A101164 From _G. C. Greubel_, Sep 17 2021: (Start)
%F A101164 T(n, k) = Sum_{j=0..n-k} binomial(n-k, j)*binomial(k, j)*2^j - binomial(n,k).
%F A101164 T(n, 1) = n-1, n > 0. (End)
%e A101164 Triangle begins as:
%e A101164   0
%e A101164   0, 0;
%e A101164   0, 1,  0;
%e A101164   0, 2,  2,   0;
%e A101164   0, 3,  7,   3,   0;
%e A101164   0, 4, 15,  15,   4,   0;
%e A101164   0, 5, 26,  43,  26,   5,  0;
%e A101164   0, 6, 40,  94,  94,  40,  6, 0;
%e A101164   0, 7, 57, 175, 251, 175, 57, 7, 0;
%t A101164 T[n_, k_]:= Hypergeometric2F1[-k, k-n, 1, 2] - Binomial[n, k];
%t A101164 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 17 2021 *)
%o A101164 (Haskell)
%o A101164 a101164 n k = a101164_tabl !! n !! k
%o A101164 a101164_row n = a101164_tabl !! n
%o A101164 a101164_tabl = zipWith (zipWith (-)) a008288_tabl a007318_tabl
%o A101164 -- _Reinhard Zumkeller_, Jul 30 2013
%o A101164 (Magma)
%o A101164 A101164:= func< n,k | (&+[Binomial(n-k,j)*Binomial(k,j)*2^j: j in [0..n-k]]) - Binomial(n,k) >;
%o A101164 [A101164(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 17 2021
%o A101164 (Sage)
%o A101164 def T(n,k): return simplify(hypergeometric([-n+k, -k], [1], 2)) - binomial(n,k)
%o A101164 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 17 2021
%Y A101164 Cf. A001477, A005449, A008288, A094706, A101165, A101166, A225413.
%K A101164 nonn,tabl
%O A101164 0,8
%A A101164 _Reinhard Zumkeller_, Dec 03 2004