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A176291 A symmetrical triangle based on Narayana numbers and Eulerian numbers of type B: T(n, k) = 2 + A060187(n, k) - 2*binomial(n, k)*binomial(n+1, k)/(k+1).

%I A176291 #8 Sep 08 2022 08:45:52
%S A176291 1,1,1,1,2,1,1,13,13,1,1,58,192,58,1,1,209,1584,1584,209,1,1,682,
%T A176291 10335,23200,10335,682,1,1,2125,60267,258745,258745,60267,2125,1,1,
%U A176291 6482,330942,2482938,4671488,2482938,330942,6482,1,1,19585,1755262,21702934,69402712,69402712,21702934,1755262,19585,1
%N A176291 A symmetrical triangle based on Narayana numbers and Eulerian numbers of type B: T(n, k) = 2 + A060187(n, k) - 2*binomial(n, k)*binomial(n+1, k)/(k+1).
%C A176291 Row sums are: {1, 2, 4, 28, 310, 3588, 45236, 642276, 10312214, 185760988, 3715773650, ...}.
%H A176291 G. C. Greubel, <a href="/A176291/b176291.txt">Rows n = 0..100 of triangle, flattened</a>
%F A176291 T(n, k) = 2 + A060187(n, k) - 2*binomial(n, k)*binomial(n+1, k)/(k+1), where A060187(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)* (2*j+1)^n.
%e A176291 Triangle begins as:
%e A176291   1;
%e A176291   1,    1;
%e A176291   1,    2,      1;
%e A176291   1,   13,     13,       1;
%e A176291   1,   58,    192,      58,       1;
%e A176291   1,  209,   1584,    1584,     209,       1;
%e A176291   1,  682,  10335,   23200,   10335,     682,      1;
%e A176291   1, 2125,  60267,  258745,  258745,   60267,   2125,    1;
%e A176291   1, 6482, 330942, 2482938, 4671488, 2482938, 330942, 6482, 1;
%p A176291 b:= binomial; T:= 2 + sum((-1)^j*b(n+1,j)*(2*(k-j)+1)^n, j=0..k) - 2*b(n, k)*b(n+1, k)/(k+1); seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10); # _G. C. Greubel_, Nov 23 2019
%t A176291 (* First program *)
%t A176291 p[x_, n_]= (1 - x)^(n+1)*Sum[(2*k+1)^n*x^k, {k, 0, Infinity}];(*A060187*)
%t A176291 f[n_, m_]:= CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m+1]];
%t A176291 T[n_, m_]:= 2 -(-f[n, m] +2*Binomial[n, m]*Binomial[n+1, m]/(m+1));
%t A176291 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%t A176291 (* Second program *)
%t A176291 B:=Binomial; T[n_,k_]:= T[n,k]= 2 +Sum[(-1)^j*B[n+1,j]*(2*(k-j)+1)^n, {j, 0,k}] -2*B[n,k]*B[n+1,k]/(k+1); Table[T[n,k], {n,0,10}, {k,0,n} ]//Flatten (* _G. C. Greubel_, Nov 23 2019 *)
%o A176291 (PARI) T(n,k) = b=binomial; 2 + sum(j=0,k, (-1)^j*b(n+1,j)*(2*(k-j)+1)^n) - 2*b(n, k)*b(n+1, k)/(k+1); \\ _G. C. Greubel_, Nov 23 2019
%o A176291 (Magma) B:=Binomial; [2*(1 - B(n,k)*B(n+1,k)/(k+1)) + (&+[(-1)^j*B(n+1,j) *(2*(k-j)+1)^n: j in [0..k]]): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 23 2019
%o A176291 (Sage) b=binomial; [[2 + sum( (-1)^j*b(n+1,j)*(2*(k-j)+1)^n for j in (0..k)) - 2*b(n, k)*b(n+1, k)/(k+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 23 2019
%o A176291 (GAP) B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> 2*(1 - B(n,k)*B(n+1,k)/(k+1)) + Sum([0..k], j-> (-1)^j*B(n+1,j)*(2*(k-j)+1)^n) ))); # _G. C. Greubel_, Nov 23 2019
%Y A176291 Cf. A007318, A060187.
%K A176291 nonn,tabl
%O A176291 0,5
%A A176291 _Roger L. Bagula_, Apr 14 2010
%E A176291 Edited by _G. C. Greubel_, Nov 23 2019