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).

1, 1, 1, 1, 2, 1, 1, 13, 13, 1, 1, 58, 192, 58, 1, 1, 209, 1584, 1584, 209, 1, 1, 682, 10335, 23200, 10335, 682, 1, 1, 2125, 60267, 258745, 258745, 60267, 2125, 1, 1, 6482, 330942, 2482938, 4671488, 2482938, 330942, 6482, 1, 1, 19585, 1755262, 21702934, 69402712, 69402712, 21702934, 1755262, 19585, 1

Offset: 0

Author

Roger L. Bagula, Apr 14 2010

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 4, 28, 310, 3588, 45236, 642276, 10312214, 185760988, 3715773650, ...}.

Examples

			Triangle begins as:
  1;
  1,    1;
  1,    2,      1;
  1,   13,     13,       1;
  1,   58,    192,      58,       1;
  1,  209,   1584,    1584,     209,       1;
  1,  682,  10335,   23200,   10335,     682,      1;
  1, 2125,  60267,  258745,  258745,   60267,   2125,    1;
  1, 6482, 330942, 2482938, 4671488, 2482938, 330942, 6482, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A007318, A060187.

Programs

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

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

Maple

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

Mathematica

(* First program *)
p[x_, n_]= (1 - x)^(n+1)*Sum[(2*k+1)^n*x^k, {k, 0, Infinity}];(*A060187*)
f[n_, m_]:= CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m+1]];
T[n_, m_]:= 2 -(-f[n, m] +2*Binomial[n, m]*Binomial[n+1, m]/(m+1));
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
(* Second program *)
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 *)

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

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

Formula

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.

Extensions

Edited by G. C. Greubel, Nov 23 2019