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A176487 Triangle read by rows: T(n,k) = binomial(n,k) + A008292(n+1,k+1) - 1.

%I A176487 #14 Jan 04 2025 22:47:49
%S A176487 1,1,1,1,5,1,1,13,13,1,1,29,71,29,1,1,61,311,311,61,1,1,125,1205,2435,
%T A176487 1205,125,1,1,253,4313,15653,15653,4313,253,1,1,509,14635,88289,
%U A176487 156259,88289,14635,509,1,1,1021,47875,455275,1310479,1310479,455275,47875,1021,1
%N A176487 Triangle read by rows: T(n,k) = binomial(n,k) + A008292(n+1,k+1) - 1.
%H A176487 G. C. Greubel, <a href="/A176487/b176487.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A176487 T(n, k) = A007318(n,k) + A008292(n+1,k+1) - 1, 0 <= k <= n.
%F A176487 Sum_{k=0..n} T(n, k) = 2^n - n + A033312(n+1) (row sums).
%F A176487 T(n, k) = 2*A141689(n+1,k+1) - 1. - _R. J. Mathar_, Jan 19 2011
%F A176487 From _G. C. Greubel_, Dec 31 2024: (Start)
%F A176487 T(n, n-k) = T(n, k).
%F A176487 T(n, 1) = A036563(n+1).
%F A176487 Sum_{k=0..n} (-1)^k * T(n,k) = ((-1)^(n/2)*A000182(n/2 + 1) - 1)*(1 + (-1)^n)/2 + [n=0]. (End)
%e A176487 Triangle begins as:
%e A176487   1;
%e A176487   1,    1;
%e A176487   1,    5,     1;
%e A176487   1,   13,    13,      1;
%e A176487   1,   29,    71,     29,       1;
%e A176487   1,   61,   311,    311,      61,       1;
%e A176487   1,  125,  1205,   2435,    1205,     125,      1;
%e A176487   1,  253,  4313,  15653,   15653,    4313,    253,     1;
%e A176487   1,  509, 14635,  88289,  156259,   88289,  14635,   509,    1;
%e A176487   1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021,   1;
%p A176487 A176487 := proc(n,k)
%p A176487     binomial(n,k)+A008292(n+1,k+1)-1 ;
%p A176487 end proc: # _R. J. Mathar_, Jun 16 2015
%t A176487 Needs["Combinatorica`"];
%t A176487 T[n_, k_, 0]:= Binomial[n, k];
%t A176487 T[n_, k_, 1]:= Eulerian[1 + n, k];
%t A176487 T[n_, k_, q_]:= T[n,k,q] = T[n,k,q-1] + T[n,k,q-2] - 1;
%t A176487 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten
%o A176487 (Magma)
%o A176487 A176487:= func< n, k | Binomial(n, k) + EulerianNumber(n+1, k) - 1 >;
%o A176487 [A176487(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 31 2024
%o A176487 (SageMath)
%o A176487 # from sage.all import * # (use for Python)
%o A176487 from sage.combinat.combinat import eulerian_number
%o A176487 def A176487(n,k): return binomial(n,k) +eulerian_number(n+1,k) -1
%o A176487 print(flatten([[A176487(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Dec 31 2024
%Y A176487 Cf. A000182, A007318, A008292, A033312, A036563, A141689.
%K A176487 nonn,tabl,easy
%O A176487 0,5
%A A176487 _Roger L. Bagula_, Apr 19 2010