A154979 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=2, read by rows.

1, 1, 1, 1, 10, 1, 1, 27, 27, 1, 1, 60, 374, 60, 1, 1, 125, 2162, 2162, 125, 1, 1, 254, 9967, 52196, 9967, 254, 1, 1, 511, 42221, 615635, 615635, 42221, 511, 1, 1, 1024, 172780, 5760960, 27955622, 5760960, 172780, 1024, 1, 1, 2049, 697068, 49168044, 664126822, 664126822, 49168044, 697068, 2049, 1

Offset

0,5

Comments

Row sums are: {1, 2, 12, 56, 496, 4576, 72640, 1316736, 39825152, 1427987968, 84417887232, ...}.

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=2.

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=2. - G. C. Greubel, Mar 01 2021

Example

Triangle begins as:
  1;
  1,    1;
  1,   10,      1;
  1,   27,     27,        1;
  1,   60,    374,       60,         1;
  1,  125,   2162,     2162,       125,         1;
  1,  254,   9967,    52196,      9967,       254,        1;
  1,  511,  42221,   615635,    615635,     42221,      511,      1;
  1, 1024, 172780,  5760960,  27955622,   5760960,   172780,   1024,    1;
  1, 2049, 697068, 49168044, 664126822, 664126822, 49168044, 697068, 2049, 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, 2]], 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, 2], {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, 2) 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, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 01 2021

Crossrefs

Cf. A154982 (m=0), A154980 (m=1), this sequence (m=3).

Sequence in context: A166341 A113280 A159041 * A146765 A190152 A154984

Adjacent sequences: A154976 A154977 A154978 * A154980 A154981 A154982

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 18 2009

Extensions

Edited by G. C. Greubel, Mar 01 2021

Status

approved