A174912 Triangle read by rows: T(n, m) = 1 + (binomial(n, m) - Eulerian(n+1, m))^2.

1, 1, 1, 1, 5, 1, 1, 65, 65, 1, 1, 485, 3601, 485, 1, 1, 2705, 85265, 85265, 2705, 1, 1, 12997, 1382977, 5740817, 1382977, 12997, 1, 1, 57601, 18249985, 242861057, 242861057, 18249985, 57601, 1, 1, 244037, 212576401, 7775359685, 24373454401, 7775359685, 212576401, 244037, 1

Offset

0,5

Comments

Row sums are: {1, 2, 7, 132, 4573, 175942, 8532767, 522337288, 40349814649, 3852213868170, 446653836767587, ...}.

Links

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

Formula

T(n, m) = 1 + (binomial(n, m) - Eulerian(n+1, m))^2, where Eulerian(n,k) = A008292(n,k).

Example

Triangle begins as:
  1;
  1,     1;
  1,     5,       1;
  1,    65,      65,       1;
  1,   485,    3601,     485,       1;
  1,  2705,   85265,   85265,    2705,     1;
  1, 12997, 1382977, 5740817, 1382977, 12997, 1;

Mathematica

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

Prog

(PARI) Eulerian(n, k) = sum(j=0, k+1, (-1)^j*binomial(n+1, j)*(k-j+1)^n);
{T(n, k) = 1 + (binomial(n, k) - Eulerian(n+1, k))^2 };
for(n=0, 10, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Apr 25 2019
(Magma) Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[[1 + (Binomial(n, k) - Eulerian(n+1, k))^2: k in [0..n]]: n in [0..12]]; // G. C. Greubel, Apr 25 2019
(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): return 1 + (binomial(n, k) - Eulerian(n+1, k))^2
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 25 2019

Crossrefs

Cf. A008292.

Sequence in context: A156691 A246051 A111820 * A106238 A173475 A174919

Adjacent sequences: A174909 A174910 A174911 * A174913 A174914 A174915

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 02 2010

Extensions

Edited by G. C. Greubel, Apr 25 2019

Status

approved