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A154983 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) + 2^(n-2)*[n>=3])*x*p(x, n-2, m) and m=0, read by rows. 4
1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 24, 70, 24, 1, 1, 49, 358, 358, 49, 1, 1, 98, 1559, 4076, 1559, 98, 1, 1, 195, 6361, 40003, 40003, 6361, 195, 1, 1, 388, 25372, 345692, 862598, 345692, 25372, 388, 1, 1, 773, 100640, 2813688, 16569442, 16569442, 2813688, 100640, 773, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 6, 24, 120, 816, 7392, 93120, 1605504, 38969088, 1310965248, ...}.

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) + 2^(n-2)*[n>=3])*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) + 2^(n-2)*[n>=3])*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,  11,     11,       1;

  1,  24,     70,      24,        1;

  1,  49,    358,     358,       49,        1;

  1,  98,   1559,    4076,     1559,       98,       1;

  1, 195,   6361,   40003,    40003,     6361,     195,      1;

  1, 388,  25372,  345692,   862598,   345692,   25372,    388,   1;

  1, 773, 100640, 2813688, 16569442, 16569442, 2813688, 100640, 773, 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] + Boole[n>=3]*2^(n-2)*x*p[x, n-2, m] ];

Table[CoefficientList[ExpandAll[p[x, n, 0]], x], {n, 0, 10}]//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^(m+n-1) + Boole[n>=3]*2^(n-2))*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

    elif (n<3): return T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m)

    else: return T(n-1, k, m) + T(n-1, k-1, m) + (2^(n+m-1) +2^(n-2))*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;

  elif (n lt 3) then return T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m);

  else return T(n-1, k, m) + T(n-1, k-1, m) + (2^(n+m-1)+2^(n-2))*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), A154984 (m=1), A154985 (m=3).

Cf. A154979, A154980, A154982, A154986.

Sequence in context: A146898 A152970 A154986 * A324916 A156534 A168287

Adjacent sequences:  A154980 A154981 A154982 * A154984 A154985 A154986

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 December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)