A327917 Triangle T read by rows: T(k, n) = A(k-n, k) with the array A(k, n) = F(2*k+n) = A000045(2*k+n), for k >= 0 and n >= 0.
0, 1, 1, 3, 2, 1, 8, 5, 3, 2, 21, 13, 8, 5, 3, 55, 34, 21, 13, 8, 5, 144, 89, 55, 34, 21, 13, 8, 377, 233, 144, 89, 55, 34, 21, 13, 987, 610, 377, 233, 144, 89, 55, 34, 21, 2584, 1597, 987, 610, 377, 233, 144, 89, 55, 34, 6765, 4181, 2584, 1597, 987, 610, 377, 233, 144, 89, 55
Offset: 0
Examples
The Array A(k, n) begins: k\n 0 1 2 3 4 5 ... ----------------------------- 0: 0 1 1 2 3 5 ... F(n) 1: 1 2 3 5 8 13 ... F(n+2) 2: 3 5 8 13 21 34 ... F(n+4) 3: 8 13 21 34 55 89 ... F(n+6) 4: 21 34 55 89 144 233 ... F(n+8) 5: 55 89 144 233 377 610 ... F(n+10) ... --------------------------------------- The triangle T(k, n) begins: k\n 0 1 2 3 4 5 6 7 8 9 10 ... ------------------------------------------------------ 0: 0 1: 1 1 2: 3 2 1 3: 8 5 3 2 4: 21 13 8 5 3 5: 55 34 21 13 8 5 6: 144 89 55 34 21 13 8 7: 377 233 144 89 55 34 21 13 8: 987 610 377 233 144 89 55 34 21 9: 2584 1597 987 610 377 233 144 89 55 34 10: 6765 4181 2584 1597 987 610 377 233 144 89 55 ...
Crossrefs
Formula
A(k, n) = Sum_{j=0..k} binomial(k, j)*F(n+j) = F(2*k+n), for k >= 0 and n >= 0.
T(k, n) = A(k - n, n) = F(2*k - n), for k >= 0 and n = 0..k, with the Fibonacci numbers F = A000045.
Recurrence: T(k,0) = F(2*k), k >= 0, T(k, n) = T(k, n-1) - T(k-1, n-1), k >= 1, n = 1..k, and T(k, n) = 0 if k < n.
O.g.f. for row polynomials R(n, x) = Sum_{n=0..k} T(k, n)*x^n:
G(x, z) = Sum_{n=0} R(n, x)*z^n = z*(1 + x - 2*x*z)/((1 - 3*z + z^2)*(1 - x*z - (x*z)^2)).
T(k, 0) = Sum_{n=0..k} binomial(k,n)*T(n, n), k >= 0 (binomial transform).
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