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A178346 Triangle read by rows: T(n, k, m) = binomial(n, k) - m*binomial(n, k)*binomial(n+1, k)/(k+1) + m*A008292(n+1, k+1) with m = 3.

%I A178346 #12 Oct 06 2024 09:44:37
%S A178346 1,1,1,1,5,1,1,18,18,1,1,52,144,52,1,1,131,766,766,131,1,1,303,3273,
%T A178346 6743,3273,303,1,1,664,12312,45422,45422,12312,664,1,1,1406,42844,
%U A178346 261230,463348,261230,42844,1406,1,1,2913,141936,1358100,3915312,3915312,1358100,141936,2913,1,1,5953,455481,6595734,29172972,47114784,29172972,6595734,455481,5953,1
%N A178346 Triangle read by rows: T(n, k, m) = binomial(n, k) - m*binomial(n, k)*binomial(n+1, k)/(k+1) + m*A008292(n+1, k+1) with m = 3.
%H A178346 G. C. Greubel, <a href="/A178346/b178346.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A178346 T(n, k, m) = binomial(n, k) - m*binomial(n, k)*binomial(n+1, k)/(k+1) + m*Eulerian(n+1, k+1) with m = 3, and Eulerian(n,k) = A008292(n,k).
%F A178346 Sum_{k=0..n} T(n, k) = 2^n + 3*(n+1)! - 3*Catalan(n+1) = 2^n + 3*A056986(n+1). - _G. C. Greubel_, Oct 05 2024
%e A178346 Triangle begins as:
%e A178346   1;
%e A178346   1,    1;
%e A178346   1,    5,      1;
%e A178346   1,   18,     18,       1;
%e A178346   1,   52,    144,      52,       1;
%e A178346   1,  131,    766,     766,     131,       1;
%e A178346   1,  303,   3273,    6743,    3273,     303,       1;
%e A178346   1,  664,  12312,   45422,   45422,   12312,     664,      1;
%e A178346   1, 1406,  42844,  261230,  463348,  261230,   42844,   1406,    1;
%e A178346   1, 2913, 141936, 1358100, 3915312, 3915312, 1358100, 141936, 2913,   1;
%t A178346 EulerianNumber[n_, k_] := EulerianNumber[n, k] = Sum[(-1)^j*(k-j)^n*Binomial[n+ 1, j], {j,0,k}];
%t A178346 A178346[n_, k_, m_]:= Binomial[n, k] - m*Binomial[n, k]*Binomial[n+1, k]/(k+1) + m*EulerianNumber[n+1, k+1];
%t A178346 Table[A178346[n,k,3], {n,0,15}, {k,0,n}]//Flatten
%o A178346 (Magma)
%o A178346 A178346:= func< n,k | Binomial(n, k) - 3*(Binomial(n, k)*Binomial(n+1, k)/(k+1)) + 3*EulerianNumber(n+1, k) >;
%o A178346 [A178346(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Oct 05 2024
%o A178346 (SageMath)
%o A178346 def A008292(n,k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in (0..k))
%o A178346 def A178346(n,k): return binomial(n,k) - 3*binomial(n,k)*binomial(n+1,k)/(k+1) + 3*A008292(n+1,k+1)
%o A178346 flatten([[A178346(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Oct 05 2024
%Y A178346 Cf. A008292.
%K A178346 nonn,tabl
%O A178346 0,5
%A A178346 _Roger L. Bagula_, May 25 2010
%E A178346 Edited by _G. C. Greubel_, Oct 05 2024