A176487 - OEIS

%I #14 Jan 04 2025 22:47:49

%S 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 1205,125,1,1,253,4313,15653,15653,4313,253,1,1,509,14635,88289,

%U 156259,88289,14635,509,1,1,1021,47875,455275,1310479,1310479,455275,47875,1021,1

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

%H G. C. Greubel, <a href="/A176487/b176487.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = A007318(n,k) + A008292(n+1,k+1) - 1, 0 <= k <= n.

%F Sum_{k=0..n} T(n, k) = 2^n - n + A033312(n+1) (row sums).

%F T(n, k) = 2*A141689(n+1,k+1) - 1. - _R. J. Mathar_, Jan 19 2011

%F From _G. C. Greubel_, Dec 31 2024: (Start)

%F T(n, n-k) = T(n, k).

%F T(n, 1) = A036563(n+1).

%F 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 Triangle begins as:

%e   1;

%e   1,    1;

%e   1,    5,     1;

%e   1,   13,    13,      1;

%e   1,   29,    71,     29,       1;

%e   1,   61,   311,    311,      61,       1;

%e   1,  125,  1205,   2435,    1205,     125,      1;

%e   1,  253,  4313,  15653,   15653,    4313,    253,     1;

%e   1,  509, 14635,  88289,  156259,   88289,  14635,   509,    1;

%e   1, 1021, 47875, 455275, 1310479, 1310479, 455275, 47875, 1021,   1;

%p A176487 := proc(n,k)

%p     binomial(n,k)+A008292(n+1,k+1)-1 ;

%p end proc: # _R. J. Mathar_, Jun 16 2015

%t Needs["Combinatorica`"];

%t T[n_, k_, 0]:= Binomial[n, k];

%t T[n_, k_, 1]:= Eulerian[1 + n, k];

%t T[n_, k_, q_]:= T[n,k,q] = T[n,k,q-1] + T[n,k,q-2] - 1;

%t Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten

%o (Magma)

%o A176487:= func< n, k | Binomial(n, k) + EulerianNumber(n+1, k) - 1 >;

%o [A176487(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 31 2024

%o (SageMath)

%o # from sage.all import * # (use for Python)

%o from sage.combinat.combinat import eulerian_number

%o def A176487(n,k): return binomial(n,k) +eulerian_number(n+1,k) -1

%o print(flatten([[A176487(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Dec 31 2024

%Y Cf. A000182, A007318, A008292, A033312, A036563, A141689.

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Apr 19 2010