A060920 Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).

1, 2, 1, 5, 5, 1, 13, 20, 9, 1, 34, 71, 51, 14, 1, 89, 235, 233, 105, 20, 1, 233, 744, 942, 594, 190, 27, 1, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 1597, 6865, 12473, 12402, 7285, 2534, 490, 44, 1, 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1

Offset

0,2

Comments

Companion triangle (odd-indexed members) A060921.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.

Formula

T(n, k) = A037027(2*n-k, k).

T(n, k) = ((2*(n-k) + 1)*A060921(n-1, k-1) + 4*n*T(n-1, k-1))/(5*k), n >= k >= 1.

T(n, 0) = F(n)^2 + F(n+1)^2 = A001519(n), with the Fibonacci numbers F(n) = A000045(n).

Sum_{k=0..n} T(n, k) = (2^(2*n + 1) + 1)/3 = A007583(n).

G.f. for column m >= 0: x^m*pFe(m+1, x)/(1-3*x+x^2)^(m+1), where pFe(n, x) := Sum_{m=0..n} A061176(n, m)*x^m (row polynomials of signed triangle A061176).

G.f.: (1-x*(1+y))/(1 - (3+2*y)*x + (1+y)^2*x^2). - Vladeta Jovovic, Oct 11 2003

Example

Triangle begins as:
      1;
      2,     1;
      5,     5,      1;
     13,    20,      9,      1;
     34,    71,     51,     14,      1;
     89,   235,    233,    105,     20,     1;
    233,   744,    942,    594,    190,    27,     1;
    610,  2285,   3522,   2860,   1295,   315,    35,    1;
   1597,  6865,  12473,  12402,   7285,  2534,   490,   44,    1;
   4181, 20284,  42447,  49963,  36122, 16407,  4578,  726,   54,  1;
  10946, 59155, 140109, 190570, 163730, 91959, 33705, 7776, 1035, 65, 1;

Mathematica

A060920[n_, k_]:= Sum[Binomial[2*n-k-j, j]*Binomial[2*n-k-2*j, k], {j, 0, 2*n-k}];
Table[A060920[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

Prog

(Magma)
A060920:= func< n, k | (&+[Binomial(2*n-k-j, j)*Binomial(2*n-k-2*j, k): j in [0..2*n-k]]) >;
[A060920(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2021
(Sage)
def A060920(n, k): return sum(binomial(2*n-k-j, j)*binomial(2*n-k-2*j, k) for j in (0..2*n-k))
flatten([[A060920(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2021

Crossrefs

Column sequences: A001519 (k=0), A054444 (k=1), A061178 (k=2), A061179 (k=3), A061180 (k=4), A061181 (k=5).

Cf. A000045, A001519, A007583, A037027, A060921, A061176.

Sequence in context: A209148 A126124 A123971 * A107842 A126216 A124733

Adjacent sequences: A060917 A060918 A060919 * A060921 A060922 A060923

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Apr 20 2001

Status

approved