A172368 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c is a sequence defined in comments.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 5, 15, 15, 15, 5, 1, 1, 7, 35, 105, 105, 35, 7, 1, 1, 9, 63, 315, 945, 315, 63, 9, 1, 1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1, 1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 1, 1; 1, 1, 1, 1; 1, 1, 1, 1, 1; 1, 3, 3, 3, 3, 1; 1, 5, 15, 15, 15, 5, 1; 1, 7, 35, 105, 105, 35, 7, 1; 1, 9, 63, 315, 945, 315, 63, 9, 1; 1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1; 1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Mathematica
f[n_, q_]:= f[n, q]= If[n==0,0,If[n<4, 1, f[n-1, q] + q*f[n-4, q]]]; c[n_, q_]:= Product[f[j, q], {j,n}]; T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])]; Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 08 2021 *) -
Sage
@CachedFunction def f(n,q): return 0 if (n==0) else 1 if (n<4) else f(n-1, q) + q*f(n-4, q) def c(n,q): return product( f(j,q) for j in (1..n) ) def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q))) flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021
Formula
T(n, k, q) = round( c(n,q)/(c(k,q)*c(n-k,q)) ), where c(n, q) = Product_{j=1..n} f(j, q), f(n, q) = f(n-1, q) + q*f(n-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 2. - G. C. Greubel, May 08 2021
Extensions
Definition corrected to give integral terms, G. C. Greubel, May 08 2021
Comments