A174728 Triangle read by rows: T(n, m, q) = (1-q^n)*Eulerian(n+1, m) - (1-q^n) + 1, with q = 2.

1, 1, 1, 1, -8, 1, 1, -69, -69, 1, 1, -374, -974, -374, 1, 1, -1735, -9330, -9330, -1735, 1, 1, -7496, -74969, -152144, -74969, -7496, 1, 1, -31241, -545083, -1983485, -1983485, -545083, -31241, 1, 1, -127754, -3724784, -22499414, -39828194, -22499414, -3724784, -127754, 1

Offset

0,5

Comments

Row sums are: {1, 2, -6, -136, -1720, -22128, -317072, -5119616, -92532096, -1854311680, -40834875136, ...}.

Links

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

Formula

T(n, m, q) = (1 - q^n)*Eulerian(n + 1, m) - (1 - q^n) + 1, where q = 2.

Example

Triangle begins as:
  1;
  1,      1;
  1,     -8,       1;
  1,    -69,     -69,        1;
  1,   -374,    -974,     -374,        1;
  1,  -1735,   -9330,    -9330,    -1735,       1;
  1,  -7496,  -74969,  -152144,   -74969,   -7496,      1;
  1, -31241, -545083, -1983485, -1983485, -545083, -31241,  1;

Mathematica

Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
With[{q = 2}, Table[(1-q^n)*(Eulerian[n+1, m]-1)+1, {n, 0, 10}, {m, 0, n}] ]//Flatten (* G. C. Greubel, Apr 20 2019 *)

Prog

(PARI) q=2; {eulerian(n, k) = sum(j=0, k+1, (-1)^j*binomial(n+1, j)*(k-j+1)^n)};
for(n=0, 10, for(k=0, n, print1((1-q^n)*(eulerian(n+1, k)-1)+1, ", "))) \\ G. C. Greubel, Apr 20 2019
(Magma) q:=2; Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >; [[(1-q^n)*(Eulerian(n+1, k)-1) +1: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 20 2019
(Sage)
q=2;
def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k+1))
[[(1-q^n)*(Eulerian(n+1, k)-1)+1 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 20 2019

Crossrefs

Sequence in context: A176642 A172346 A178048 * A015121 A156766 A178046

Adjacent sequences: A174725 A174726 A174727 * A174729 A174730 A174731

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 28 2010

Extensions

Edited by G. C. Greubel, Apr 20 2019

Status

approved