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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 18 10:28 EDT 2022. Contains 353807 sequences. (Running on oeis4.)