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A154986 Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows. 4
1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 24, 70, 24, 1, 1, 45, 314, 314, 45, 1, 1, 76, 1079, 2728, 1079, 76, 1, 1, 119, 3045, 16995, 16995, 3045, 119, 1, 1, 176, 7420, 80464, 186758, 80464, 7420, 176, 1, 1, 249, 16164, 307124, 1490862, 1490862, 307124, 16164, 249, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The sequence is row sum dual to the Eulerian numbers A008292.

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2).

From G. C. Greubel, Mar 01 2021: (Start)

T(n, k) = T(n-1, k) + T(n-1, k-1) + n*(n-1)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

T(n, k) = T(n, n-k).

Sum_{k=0..n} T(n, k) = n! = A000142(n). (End)

EXAMPLE

Triangle begins as:

  1;

  1,   1;

  1,   4,     1;

  1,  11,    11,      1;

  1,  24,    70,     24,       1;

  1,  45,   314,    314,      45,        1;

  1,  76,  1079,   2728,    1079,       76,       1;

  1, 119,  3045,  16995,   16995,     3045,     119,      1;

  1, 176,  7420,  80464,  186758,    80464,    7420,    176,     1;

  1, 249, 16164, 307124, 1490862,  1490862,  307124,  16164,   249,   1;

  1, 340, 32253, 991088, 9039746, 19789944, 9039746, 991088, 32253, 340, 1;

MATHEMATICA

(* First program *)

p[x_, n_]:= p[x, n, m] = If[n<2, n*x+1, (x+1)*p[x, n-1] + n*(n-1)*x*p[x, n-2]];

Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 01 2021 *)

(* Second program *)

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, T[n-1, k] +T[n-1, k-1] +n*(n-1)*T[n-2, k-1]];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 01 2021 *)

PROG

(Sage)

def T(n, k):

    if (k==0 or k==n): return 1

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

flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2021

(Magma)

function T(n, k)

  if k eq 0 or k eq n then return 1;

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

  end if; return T;

end function;

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

CROSSREFS

Cf. A000142, A009292.

Cf. A154982, A154980, A154979.

Cf. A154983, A154984, A154985.

Sequence in context: A154096 A146898 A152970 * A154983 A324916 A156534

Adjacent sequences:  A154983 A154984 A154985 * A154987 A154988 A154989

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 November 27 16:35 EST 2021. Contains 349394 sequences. (Running on oeis4.)