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

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Last modified January 24 00:05 EST 2022. Contains 350515 sequences. (Running on oeis4.)