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

2,1

Comments

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

Links

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

Formula

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

Example

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;

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A008292, A166454.

Sequence in context: A154692 A309161 A144293 * A183419 A305314 A154694

Adjacent sequences: A174095 A174096 A174097 * A174099 A174100 A174101

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 07 2010

Extensions

Edited by G. C. Greubel, Apr 25 2019

Status

approved