A154986 Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 24, 70, 24, 1, 1, 45, 314, 314, 45, 1, 1, 76, 1079, 2728, 1079, 76, 1, 1, 119, 3045, 16995, 16995, 3045, 119, 1, 1, 176, 7420, 80464, 186758, 80464, 7420, 176, 1, 1, 249, 16164, 307124, 1490862, 1490862, 307124, 16164, 249, 1

Offset

0,5

Comments

The sequence is row sum dual to the Eulerian numbers A008292.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2).

From G. C. Greubel, Mar 01 2021: (Start)

T(n, k) = T(n-1, k) + T(n-1, k-1) + n*(n-1)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

T(n, k) = T(n, n-k).

Sum_{k=0..n} T(n, k) = n! = A000142(n). (End)

Example

Triangle begins as:
  1;
  1,   1;
  1,   4,     1;
  1,  11,    11,      1;
  1,  24,    70,     24,       1;
  1,  45,   314,    314,      45,        1;
  1,  76,  1079,   2728,    1079,       76,       1;
  1, 119,  3045,  16995,   16995,     3045,     119,      1;
  1, 176,  7420,  80464,  186758,    80464,    7420,    176,     1;
  1, 249, 16164, 307124, 1490862,  1490862,  307124,  16164,   249,   1;
  1, 340, 32253, 991088, 9039746, 19789944, 9039746, 991088, 32253, 340, 1;

Mathematica

(* First program *)
p[x_, n_]:= p[x, n, m] = If[n<2, n*x+1, (x+1)*p[x, n-1] + n*(n-1)*x*p[x, n-2]];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 01 2021 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] +T[n-1, k-1] +n*(n-1)*T[n-2, k-1]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 01 2021 *)

Prog

(Sage)
def T(n, k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + n*(n-1)*T(n-2, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2021
(Magma)
function T(n, k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + n*(n-1)*T(n-2, k-1);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 01 2021

Crossrefs

Cf. A000142, A009292.

Cf. A154982, A154980, A154979.

Cf. A154983, A154984, A154985.

Sequence in context: A154096 A146898 A152970 * A154983 A324916 A156534

Adjacent sequences: A154983 A154984 A154985 * A154987 A154988 A154989

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 18 2009

Extensions

Edited by G. C. Greubel, Mar 01 2021

Status

approved