A103415 Triangle, read by rows, T(n,k) = A000129(n+1) - Sum_{j=1..k} t(n+1, j), where t(n, k) is defined in the formula section.
1, 2, 1, 5, 4, 1, 12, 11, 6, 1, 29, 28, 21, 8, 1, 70, 69, 60, 35, 10, 1, 169, 168, 157, 116, 53, 12, 1, 408, 407, 394, 333, 204, 75, 14, 1, 985, 984, 969, 884, 653, 332, 101, 16, 1, 2378, 2377, 2360, 2247, 1870, 1189, 508, 131, 18, 1, 5741, 5740, 5721, 5576, 5001, 3712, 2029, 740, 165, 20, 1
Offset: 0
Examples
Triangle begins as:
1;
2, 1;
5, 4, 1;
12, 11, 6, 1;
29, 28, 21, 8, 1;
70, 69, 60, 35, 10, 1;
169, 168, 157, 116, 53, 12, 1;
408, 407, 394, 333, 204, 75, 14, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Mathematica
t[n_, k_]:= If[k==0, (2*Boole[n<2] + LucasL[n-1, 2]*Boole[n>1])/2, Binomial[n-1, k-1]*Hypergeometric2F1[1-k, k-n, 1-n, -1]]; st[n_, k_]:= Sum[t[n+1, j], {j,k}]; T[n_, k_]:= Fibonacci[n+1, 2] - st[n, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 25 2021 *) -
PARI
Pell(n) = if( n<2, n, 2*Pell(n-1) + Pell(n-2) ); t(n, k) = if(n<3, 1, if(k==1||k==n, 1, t(n-1,k) + t(n-1,k-1) + t(n-2,k-1) )); st(n, k) = sum(i=1, k, t(n+1,i)); T(n, k) = Pell(n+1) - st(n,k); for(n=0, 10, for(k=0, n, print1(T(n,k), ",")); print()) \\ modified by G. C. Greubel, May 25 2021
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Sage
@CachedFunction def t(n,k): return 1 if (n<3) else 1 if (k==1 or k==n) else t(n-1,k) + t(n-1,k-1) + t(n-2,k-1) def st(n,k): return sum(t(n+1, j) for j in (1..k)) def T(n,k): return lucas_number1(n+1,2,-1) - st(n,k) flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021
Formula
T(n, k) = Pell(n+1) - ST(n, k), where ST(n, k) = Sum_{j=1..k} t(n+1, j), t(n, k) = t(n-1,k) + t(n-1,k-1) + t(n-2,k-1), t(n, 1) = t(n, n) = 1 and t(0, k) = t(1, k) = t(2, k) = 1.
T(n, 0) = A000129(n+1).
Sum_{k=0..n} T(n, k) = A026937(n).
From G. C. Greubel, May 25 2021: (Start)
T(n, k) = A000129(n+1) - st(n,k), where st(n, k) = Sum_{j=1..k} t(n+1, j), t(n, k) = A008288(n-1, k-1) for n >= 1 and k >= 1, and t(n, 0) = (1/2)*(2*[n<2] + A002203(n-1)*[n>1]).
T(n, n) = A000012(n).
T(n, n-1) = A005843(n+1).
T(n, n-2) = A093328(n-1).
T(n, n-3) = (4/3)*((n-3)^3 + 5*(n-3) + 3).
T(n, n-4) = (1/3)*(2*(n-4)^2 + 22*(n-4)^2 + 22*(n-4) + 39). (End)
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