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A173045 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows. 2
1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 6566, 84, 1, 1, 247, 14348916, 14348916, 247, 1, 1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1, 1, 2193, 50031545098999727, 2503155504993241601315571986085883, 2503155504993241601315571986085883, 50031545098999727, 2193, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..11 of the triangle, flattened

FORMULA

T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.

Sum_{k=0..n} T(n, k, 3) = A000295(n) + Sum_{k=0..n} 3^(n*binomial(n-2, k-1)). - G. C. Greubel, Feb 19 2021

EXAMPLE

Triangle begins as:

  1;

  1,   1;

  1,  10,            1;

  1,  29,           29,                  1;

  1,  84,         6566,                 84,            1;

  1, 247,     14348916,           14348916,          247,   1;

  1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1;

MATHEMATICA

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

Table[t[n, k, 3], {n, 0, 9}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)

PROG

(Sage)

def T(n, k, q):

    if (k==0 or k==n): return 1

    else: return binomial(n, k) -1 +q^(n*binomial(n-2, k-1))

flatten([[T(n, k, 3) for k in (0..n)] for n in (0..9)]) # G. C. Greubel, Feb 19 2021

(Magma)

T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) -1 +q^(n*Binomial(n-2, k-1)) >;

[T(n, k, 3): k in [0..n], n in [0..9]]; // G. C. Greubel, Feb 19 2021

CROSSREFS

Cf. A132044 (q=0), A007318 (q=1), A173043 (q=2), this sequence (q=3).

Cf. A000295.

Sequence in context: A190152 A154984 A173047 * A176491 A008958 A168524

Adjacent sequences:  A173042 A173043 A173044 * A173046 A173047 A173048

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 08 2010

EXTENSIONS

Edited by G. C. Greubel, Feb 19 2021

STATUS

approved

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Last modified April 20 12:58 EDT 2021. Contains 343135 sequences. (Running on oeis4.)