A176200 -id:A176200 - cp's OEIS frontend

A176204 Triangle T(n, k) = 4 * A008292(n+1, k) - 3, read by rows.

1, 1, 1, 1, 13, 1, 1, 41, 41, 1, 1, 101, 261, 101, 1, 1, 225, 1205, 1205, 225, 1, 1, 477, 4761, 9661, 4761, 477, 1, 1, 985, 17169, 62473, 62473, 17169, 985, 1, 1, 2005, 58429, 352933, 624757, 352933, 58429, 2005, 1, 1, 4049, 191357, 1820765, 5241413, 5241413, 1820765, 191357, 4049, 1

Offset: 0

Roger L. Bagula, Apr 11 2010

 This sequence belongs to the class defined by T(n, m, q) = 2*T(n, m, q-1) - 1. The first few q values gives the sequences: A008292(n+1, k) (q=0), A176200 (q=1), this sequence (q=2).
Row sums are: {1, 2, 15, 84, 465, 2862, 20139, 161256, 1451493, 14515170, 159667167, ...}.
Former title: A recursive symmetrical triangular sequence based on Eulerian numbers: q=2: T(n, m, q) = 2*T(n, m, q-1) - 1.

			Triangle begins as:
  1;
  1,    1;
  1,   13,      1;
  1,   41,     41,       1;
  1,  101,    261,     101,       1;
  1,  225,   1205,    1205,     225,       1;
  1,  477,   4761,    9661,    4761,     477,       1;
  1,  985,  17169,   62473,   62473,   17169,     985,      1;
  1, 2005,  58429,  352933,  624757,  352933,   58429,   2005,    1;
  1, 4049, 191357, 1820765, 5241413, 5241413, 1820765, 191357, 4049, 1;

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

Cf. A007318, A109128, A131061, A168625, A176200.

Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
[[4*Eulerian(n+1,k) -3: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Mar 12 2020

A008292:= (n,k) -> add((-1)^j*binomial(n+1,j)*(k-j+1)^n, j=0..k+1);
A176204:= (n,k,q) -> 2^q*( A008292(n+1,k) -1) + 1;
seq(seq( A176204(n,k,2), k=0..n), n=0..12); # G. C. Greubel, Mar 12 2020

Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j,0,k+1}];
T[n_, m_, q_]:= 2^q*Eulerian[n+1, m] - 2^q +1;
Table[T[n, m, 2], {n,0,12}, {m,0,n}]//Flatten (* modified by G. C. Greubel, Mar 12 2020 *)

Eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n);
T(n,k,q) = 2^q*Eulerian(n+1,k) - (2^q - 1); \\ G. C. Greubel, Mar 12 2020

def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
def T(n,k,q): return 2^q*Eulerian(n+1,k) - 2^q + 1
[[T(n,k,2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

T(n, m, q) = 2*T(n, m, q-1) - 1, with T(n, m, 0) = A008292(n+1, m).
From G. C. Greubel, Mar 12 2020: (Start)
T(n, k, q) = 2^q * A008292(n+1, k) - (2^q - 1).
Sum_{k=0..n} T(n, k, q) = (n+1)*( 2^q * n! - 2^q + 1) (row sums). (End)

Edited by G. C. Greubel, Mar 12 2020

A174033 T(n, m) = Sum_{i=0..10} floor(Eulerian(n+1, m)/2^i).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 49, 130, 49, 1, 1, 110, 599, 599, 110, 1, 1, 236, 2376, 4826, 2376, 236, 1, 1, 487, 8578, 31220, 31220, 8578, 487, 1, 1, 997, 29200, 176378, 312223, 176378, 29200, 997, 1, 1, 2018, 95630, 909937, 2619425, 2619425, 909937, 95630, 2018, 1

Offset: 0

Roger L. Bagula, Mar 06 2010

			Triangle begins:
  {1},
  {1, 1},
  {1, 7, 1},
  {1, 19, 19, 1},
  {1, 49, 130, 49, 1},
  {1, 110, 599, 599, 110, 1},
  {1, 236, 2376, 4826, 2376, 236, 1},
  {1, 487, 8578, 31220, 31220, 8578, 487, 1},
  {1, 997, 29200, 176378, 312223, 176378, 29200, 997, 1},
  {1, 2018, 95630, 909937, 2619425, 2619425, 909937, 95630, 2018, 1},
  ...

Cf. A008292, A176200.

<< DiscreteMath`Combinatorica`
Sum[Floor[Eulerian[n + 1, m]/2^i], {i, 0, 10}];
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

T(n, m) = Sum_{i=0..10} floor(Eulerian(n+1, m)/2^i), where Eulerian(n, k) = A008292(n, k).

cp's OEIS Frontend

A176204 Triangle T(n, k) = 4 * A008292(n+1, k) - 3, read by rows.

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