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A173120
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Triangle T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = -4, read by rows.
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6
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1, 1, 1, 1, -2, 1, 1, -1, -1, 1, 1, 0, -2, 0, 1, 1, 1, 14, 14, 1, 1, 1, 2, 15, 28, 15, 2, 1, 1, 3, 17, -21, -21, 17, 3, 1, 1, 4, 20, -4, -42, -4, 20, 4, 1, 1, 5, 24, 16, 210, 210, 16, 24, 5, 1, 1, 6, 29, 40, 226, 420, 226, 40, 29, 6, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k, q) = q*[n=2] + Sum_{j=0..5} q^j*binomial(n-2*j, k-j)*[n>2*j] with T(n,0) = T(n,n) = 1 for q = -4.
Sum_{k=0..n} T(n, k, q) = [n=0] + q*[n=2] + Sum_{j=0..5} q^j*2^(n-2*j)*[n > 2*j] for q = -4. - G. C. Greubel, Apr 27 2021
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, -2, 1;
1, -1, -1, 1;
1, 0, -2, 0, 1;
1, 1, 14, 14, 1, 1;
1, 2, 15, 28, 15, 2, 1;
1, 3, 17, -21, -21, 17, 3, 1;
1, 4, 20, -4, -42, -4, 20, 4, 1;
1, 5, 24, 16, 210, 210, 16, 24, 5, 1;
1, 6, 29, 40, 226, 420, 226, 40, 29, 6, 1;
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MATHEMATICA
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T[n_, k_, q_]:= If[k==0 || k==n, 1, q*Boole[n==2] + Sum[q^j*Binomial[n-2*j, k-j]*Boole[n>2*j], {j, 0, 5}]];
Table[T[n, k, -4], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)
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PROG
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(Sage)
@CachedFunction
def T(n, k, q): return 1 if (k==0 or k==n) else q*bool(n==2) + sum( q^j*binomial(n-2*j, k-j)*bool(n>2*j) for j in (0..5) )
flatten([[T(n, k, -4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 27 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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