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A174733 Triangle T(n, k, q) = (1-q^n)*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - (1-q^n) + 1, for q = 4, read by rows. 3
1, 1, 1, 1, -125, 1, 1, -1274, -1274, 1, 1, -9206, -19436, -9206, 1, 1, -57329, -200654, -200654, -57329, 1, 1, -327659, -1703831, -2850641, -1703831, -327659, 1, 1, -1769444, -12779324, -32046614, -32046614, -12779324, -1769444, 1, 1, -9175004, -87817904, -308018024, -462158108, -308018024, -87817904, -9175004, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

From G. C. Greubel, Feb 09 2021: (Start)

The triangle coefficients are connected to the Narayana numbers by T(n, k, q) = (1-q^n)*(A001263(n, k) - 1) + 1, for varying q values.

The row sums of this class of sequences, for varying q, is given by Sum_{k=1..n} T(n, k, q) = q^n * n + (1 - q^n)*C_{n}, where C_{n} are the Catalan numbers (A000108). (End)

LINKS

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

FORMULA

T(n, k, q) = (1-q^n)*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - (1-q^n) + 1, for q = 4.

From G. C. Greubel, Feb 09 2021: (Start)

T(n, k, 4) = (1-4^n)*(A001263(n,k) - 1) + 1.

Sum_{k=1..n} T(n, k, 4) = 4^n * n + (1 - 4^n)*A000108(n). (End)

EXAMPLE

Triangle begins as:

  1;

  1,        1;

  1,     -125,         1;

  1,    -1274,     -1274,         1;

  1,    -9206,    -19436,     -9206,         1;

  1,   -57329,   -200654,   -200654,    -57329,         1;

  1,  -327659,  -1703831,  -2850641,  -1703831,   -327659,        1;

  1, -1769444, -12779324, -32046614, -32046614, -12779324, -1769444, 1;

MATHEMATICA

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

Table[T[n, k, 4], {n, 12}, {k, n}]//Flatten

PROG

(Sage)

def T(n, k, q): return 1 + (1-q^n)*(1/k)*(binomial(n-1, k-1)*binomial(n, k-1) - k)

flatten([[T(n, k, 4) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021

(Magma)

T:= func< n, k, q | 1 +(1-q^n)*(1/k)*(Binomial(n-1, k-1)*Binomial(n, k-1) - k) >;

[T(n, k, 4): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021

CROSSREFS

Cf. A000108, A001263.

Cf. A000012 (q=1), A174731 (q=2), A174732 (q=3), this sequence (q=4).

Sequence in context: A009805 A227393 A005080 * A100579 A298062 A298711

Adjacent sequences:  A174730 A174731 A174732 * A174734 A174735 A174736

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Mar 28 2010

EXTENSIONS

Edited by G. C. Greubel, Feb 09 2021

STATUS

approved

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Last modified May 14 03:00 EDT 2021. Contains 343872 sequences. (Running on oeis4.)