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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
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. A000012 (q=1), A174731 (q=2), A174732 (q=3), this sequence (q=4).
Sequence in context: A009805 A227393 A005080 * A100579 A298062 A298711
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 June 18 04:26 EDT 2024. Contains 373468 sequences. (Running on oeis4.)