This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A327917 #17 Jul 09 2024 19:40:39
%S A327917 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,
%T A327917 377,233,144,89,55,34,21,13,987,610,377,233,144,89,55,34,21,2584,1597,
%U A327917 987,610,377,233,144,89,55,34,6765,4181,2584,1597,987,610,377,233,144,89,55
%N 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.
%C A327917 This is the row reversed triangle A199334.
%C A327917 This is the analog of the array and the triangle A327916, where the positive odd numbers instead of the Fibonacci numbers are used.
%C A327917 The array A arises from the following Pascal-type triangles PF(k), for k >= 0, based on A000045 (Fibonacci). For example, the Pascal type triangle PF(k), for k = 3 is
%C A327917 0 1 1 2
%C A327917 1 2 3
%C A327917 3 5
%C A327917 8
%C A327917 For k -> infinity the left-aligned row sequences build the array A(k, n), with k >= 0 and n >= 0, that is, A(k, n) = F(2*k + n). See the example section for the first rows of A.
%C A327917 The first column sequence of A, {F(2*n) = A001906(n)}_{n>=0}, is the binomial transform of the first (k=0) row sequence of A, {F(n)}_{n>=0}.
%C A327917 The triangle T is the array A read by upwards antidiagonals.
%F A327917 A(k, n) = Sum_{j=0..k} binomial(k, j)*F(n+j) = F(2*k+n), for k >= 0 and n >= 0.
%F A327917 T(k, n) = A(k - n, n) = F(2*k - n), for k >= 0 and n = 0..k, with the Fibonacci numbers F = A000045.
%F A327917 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.
%F A327917 O.g.f. for row polynomials R(n, x) = Sum_{n=0..k} T(k, n)*x^n:
%F A327917 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)).
%F A327917 T(k, 0) = Sum_{n=0..k} binomial(k,n)*T(n, n), k >= 0 (binomial transform).
%e A327917 The Array A(k, n) begins:
%e A327917 k\n 0 1 2 3 4 5 ...
%e A327917 -----------------------------
%e A327917 0: 0 1 1 2 3 5 ... F(n)
%e A327917 1: 1 2 3 5 8 13 ... F(n+2)
%e A327917 2: 3 5 8 13 21 34 ... F(n+4)
%e A327917 3: 8 13 21 34 55 89 ... F(n+6)
%e A327917 4: 21 34 55 89 144 233 ... F(n+8)
%e A327917 5: 55 89 144 233 377 610 ... F(n+10)
%e A327917 ...
%e A327917 ---------------------------------------
%e A327917 The triangle T(k, n) begins:
%e A327917 k\n 0 1 2 3 4 5 6 7 8 9 10 ...
%e A327917 ------------------------------------------------------
%e A327917 0: 0
%e A327917 1: 1 1
%e A327917 2: 3 2 1
%e A327917 3: 8 5 3 2
%e A327917 4: 21 13 8 5 3
%e A327917 5: 55 34 21 13 8 5
%e A327917 6: 144 89 55 34 21 13 8
%e A327917 7: 377 233 144 89 55 34 21 13
%e A327917 8: 987 610 377 233 144 89 55 34 21
%e A327917 9: 2584 1597 987 610 377 233 144 89 55 34
%e A327917 10: 6765 4181 2584 1597 987 610 377 233 144 89 55
%e A327917 ...
%Y A327917 Cf. A000045, A199334 (row reversed), A001906, A327916.
%Y A327917 Column sequences of T (no leading zeros) and A: from the shifted Fibonacci bisection {F(2*k) = A001906(k)} for even n, and {F(2*k+1) = A001519(k+1)}, for odd n.
%Y A327917 Row sums: 2*A094292(n+1) = F(2*(n+1)) - F(n+1), n >= 0.
%Y A327917 Alternating row sums: 2*A164267(n-1), n >= 0, with 0 for n = 0.
%K A327917 nonn,easy,tabl
%O A327917 0,4
%A A327917 _Wolfdieter Lang_, Oct 06 2019