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A154984 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=2, read by rows. 4
1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 66, 418, 66, 1, 1, 139, 2572, 2572, 139, 1, 1, 284, 12215, 65336, 12215, 284, 1, 1, 573, 52531, 818287, 818287, 52531, 573, 1, 1, 1150, 216688, 7906658, 39270110, 7906658, 216688, 1150, 1, 1, 2303, 877934, 68639058, 989843392, 989843392, 68639058, 877934, 2303, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 12, 60, 552, 5424, 90336, 1742784, 55519104, 2118725376, 132153466368, ...}.

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=2.

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

EXAMPLE

Triangle begins as:

  1;

  1,    1;

  1,   10,      1;

  1,   29,     29,        1;

  1,   66,    418,       66,         1;

  1,  139,   2572,     2572,       139,         1;

  1,  284,  12215,    65336,     12215,       284,        1;

  1,  573,  52531,   818287,    818287,     52531,      573,      1;

  1, 1150, 216688,  7906658,  39270110,   7906658,   216688,   1150,    1;

  1, 2303, 877934, 68639058, 989843392, 989843392, 68639058, 877934, 2303, 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, 2]], 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, 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

    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, 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;

  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, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 01 2021

CROSSREFS

Cf. A154983 (m=0), A154985 (m=1), this sequence (m=2).

Cf. A154979, A154980, A154982, A154986.

Sequence in context: A154979 A146765 A190152 * A173047 A173045 A176491

Adjacent sequences:  A154981 A154982 A154983 * A154985 A154986 A154987

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 17:28 EDT 2021. Contains 343996 sequences. (Running on oeis4.)