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A154982 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=0, read by rows. 7
1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 18, 50, 18, 1, 1, 35, 212, 212, 35, 1, 1, 68, 823, 2024, 823, 68, 1, 1, 133, 3131, 16415, 16415, 3131, 133, 1, 1, 262, 11968, 124890, 291902, 124890, 11968, 262, 1, 1, 519, 46278, 938394, 4619032, 4619032, 938394, 46278, 519, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 6, 20, 88, 496, 3808, 39360, 566144, 11208448, ...}.

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=0.

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=0. - G. C. Greubel, Mar 01 2021

EXAMPLE

Triangle begins as:

  1;

  1,   1;

  1,   4,     1;

  1,   9,     9,      1;

  1,  18,    50,     18,       1;

  1,  35,   212,    212,      35,       1;

  1,  68,   823,   2024,     823,      68,      1;

  1, 133,  3131,  16415,   16415,    3131,    133,     1;

  1, 262, 11968, 124890,  291902,  124890,  11968,   262,   1;

  1, 519, 46278, 938394, 4619032, 4619032, 938394, 46278, 519, 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, 0]], 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, 0], {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, 0) 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, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 01 2021

CROSSREFS

Cf. this sequence (m=0), A154980 (m=1), A154979 (m=3).

Sequence in context: A259333 A180960 A157192 * A146767 A146955 A155451

Adjacent sequences:  A154979 A154980 A154981 * A154983 A154984 A154985

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 July 28 19:33 EDT 2021. Contains 346335 sequences. (Running on oeis4.)