A154980 Triangle T(n, k, m) = coefficients of p(x, n, m) where p(x,n,m) = (x+1)*p(x, n-1, m) + 2^(m+n-1) *x*p(x, n-2, m) and m=1, read by rows.

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 32, 126, 32, 1, 1, 65, 638, 638, 65, 1, 1, 130, 2751, 9340, 2751, 130, 1, 1, 259, 11201, 93755, 93755, 11201, 259, 1, 1, 516, 44740, 809212, 2578550, 809212, 44740, 516, 1, 1, 1029, 177864, 6588864, 51390322, 51390322, 6588864, 177864, 1029, 1

Offset

0,5

Comments

Row sums are: {1, 2, 8, 32, 192, 1408, 15104, 210432, 4287488, 116316160, 4623020032, ...}.

Links

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

Formula

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

T(n, k, m) = T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m=1. - G. C. Greubel, Mar 01 2021

Example

Triangle begins as:
  1;
  1,    1;
  1,    6,      1;
  1,   15,     15,       1;
  1,   32,    126,      32,        1;
  1,   65,    638,     638,       65,        1;
  1,  130,   2751,    9340,     2751,      130,       1;
  1,  259,  11201,   93755,    93755,    11201,     259,      1;
  1,  516,  44740,  809212,  2578550,   809212,   44740,    516,    1;
  1, 1029, 177864, 6588864, 51390322, 51390322, 6588864, 177864, 1029, 1;

Mathematica

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

Prog

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

Crossrefs

Cf. A154982 (m=0), this sequence (m=1), A154979 (m=3).

Sequence in context: A295985 A086645 A168291 * A166344 A146766 A176152

Adjacent sequences: A154977 A154978 A154979 * A154981 A154982 A154983

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 18 2009

Extensions

Edited by G. C. Greubel, Mar 01 2021

Status

approved