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

0,5

Comments

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.

Links

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

Formula

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)

Example

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;

Maple

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

Mathematica

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 *)

Prog

(PARI) 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
(Magma) 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
(Sage)
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

Crossrefs

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

Sequence in context: A060361 A141596 A108477 * A176492 A174731 A342891

Adjacent sequences: A176201 A176202 A176203 * A176205 A176206 A176207

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 11 2010

Extensions

Edited by G. C. Greubel, Mar 12 2020

Status

approved