A174098 Symmetrical triangle T(n, m) = floor(Eulerian(n+1, m)/2), read by rows.

2, 5, 5, 13, 33, 13, 28, 151, 151, 28, 60, 595, 1208, 595, 60, 123, 2146, 7809, 7809, 2146, 123, 251, 7304, 44117, 78095, 44117, 7304, 251, 506, 23920, 227596, 655177, 655177, 227596, 23920, 506, 1018, 76318, 1101744, 4869057, 7862124, 4869057, 1101744, 76318, 1018

Offset: 2

Author

Roger L. Bagula, Mar 07 2010

Keywords

• nonn
• tabl

Comments

Row sums are: {2, 10, 59, 358, 2518, 20156, 181439, 1814398, 19958398, 239500796, 3113510398, 43589145596, 653837183996, ...}.

Examples

			Triangle begins as:
    2;
    5,     5;
   13,    33,     13;
   28,   151,    151,     28;
   60,   595,   1208,    595,     60;
  123,  2146,   7809,   7809,   2146,    123;
  251,  7304,  44117,  78095,  44117,   7304,   251;
  506, 23920, 227596, 655177, 655177, 227596, 23920, 506;
		

Links

• G. C. Greubel, Rows n = 2..100 of triangle, flattened

Crossrefs

Cf. A008292, A166454.

Programs

Magma

[[Floor((&+[(-1)^j*Binomial(n+2,j)*(k-j+1)^(n+1): j in [0..k+1]] )/2): k in [1..n-1]]: n in [2..12]]; // G. C. Greubel, Apr 25 2019

Mathematica

Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
Table[Floor[Eulerian[n+1, m]/2], {n, 2, 12}, {m, 1, n-1}]//Flatten (* G. C. Greubel, Apr 25 2019 *)

PARI

{T(n,k) = (sum(j=0,k+1, (-1)^j*binomial(n+2,j)*(k-j+1)^(n+1)))\2};
for(n=2,12, for(k=1,n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Apr 25 2019

Sage

[[floor(sum((-1)^j*binomial(n+2,j)*(k-j+1)^(n+1) for j in (0..k+1))/2) for k in (1..n-1)] for n in (2..12)] # G. C. Greubel, Apr 25 2019

Formula

T(n, m) = floor(Eulerian(n+1, m)/2), where Eulerian(n,k) = A008292(n,k).

Extensions

Edited by G. C. Greubel, Apr 25 2019