A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.
1, 1, 1, 1, -7, 1, 1, -51, -51, 1, 1, -239, -399, -239, 1, 1, -967, -2177, -2177, -967, 1, 1, -3639, -10191, -13831, -10191, -3639, 1, 1, -13115, -43719, -74323, -74323, -43719, -13115, 1, 1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, -7, 1; 1, -51, -51, 1; 1, -239, -399, -239, 1; 1, -967, -2177, -2177, -967, 1; 1, -3639, -10191, -13831, -10191, -3639, 1; 1, -13115, -43719, -74323, -74323, -43719, -13115, 1; 1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1;
Links
- G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Crossrefs
Programs
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Magma
T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >; [T(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021
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Mathematica
T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1); Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten
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Sage
def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1) flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021
Formula
T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=3.
Sum_{k=0..n} T(n, k, 3) = 3^n*(n+1) + 2^n*(1 - 3^n) = A027471(n+2) - A248216(n). - G. C. Greubel, Feb 09 2021
Extensions
Edited by G. C. Greubel, Feb 09 2021
Comments