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A173047 Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 3, read by rows. 4
1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 167, 84, 1, 1, 247, 738, 738, 247, 1, 1, 734, 2930, 4393, 2930, 734, 1, 1, 2193, 10955, 21904, 21904, 10955, 2193, 1, 1, 6568, 39393, 98470, 131289, 98470, 39393, 6568, 1, 1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 16 2021

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

FORMULA

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

Sum_{k=0..n} T(n, k, 3) = (1/4)*(6^n + 2^(n+2) - 4*(n-1) - 5*[n=0] - 6*[n=1]). - G. C. Greubel, Feb 16 2021

EXAMPLE

Ttiangle begins as:

  1;

  1,     1;

  1,    10,      1;

  1,    29,     29,      1;

  1,    84,    167,     84,      1;

  1,   247,    738,    738,    247,      1;

  1,   734,   2930,   4393,   2930,    734,      1;

  1,  2193,  10955,  21904,  21904,  10955,   2193,      1;

  1,  6568,  39393,  98470, 131289,  98470,  39393,   6568,     1;

  1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1;

MATHEMATICA

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

Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *)

PROG

(Sage)

def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^n*binomial(n-2, k-1) -1

flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021

(Magma)

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

[T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

CROSSREFS

Cf. A132044 (q=0), A173075 (q=1), A173046 (q=2), this sequence (q=3).

Sequence in context: A146765 A190152 A154984 * A173045 A176491 A008958

Adjacent sequences:  A173044 A173045 A173046 * A173048 A173049 A173050

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 08 2010

EXTENSIONS

Edited by G. C. Greubel, Feb 16 2021

STATUS

approved

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Last modified April 15 04:41 EDT 2021. Contains 342975 sequences. (Running on oeis4.)