OFFSET
0,8
COMMENTS
Row sums: A000045(n+1), Fibonacci numbers.
A168561*A007318 = A037027, as lower triangular matrices. Diagonal sums : A077957. - Philippe Deléham, Dec 02 2009
T(n,k) is the number of compositions of n+1 into k+1 odd parts. Example: T(4,2)=3 because we have 5 = 1+1+3 = 1+3+1 = 3+1+1.
Coefficients of monic Fibonacci polynomials (rising powers of x). Ftilde(n, x) = x*Ftilde(n-1, x) + Ftilde(n-2, x), n >=0, Ftilde(-1,x) = 0, Ftilde(0, x) = 1. G.f.: 1/(1 - x*z - z^2). Compare with Chebyshev S-polynomials (A049310). - Wolfdieter Lang, Jul 29 2014
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
J.P. Allouche and M. Mendès-France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From N. J. A. Sloane, May 10 2012
Tom Copeland, Addendum to Elliptic Lie Triad
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.
FORMULA
Sum_{k=0..n} T(n,k)*x^k = A059841(n), A000045(n+1), A000129(n+1), A006190(n+1), A001076(n+1), A052918(n), A005668(n+1), A054413(n), A041025(n), A099371(n+1), A041041(n), A049666(n+1), A041061(n), A140455(n+1), A041085(n), A154597(n+1), A041113(n) for x = 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 respectively. - Philippe Deléham, Dec 02 2009
T(2n,2k) = A085478(n,k). T(2n+1,2k+1) = A078812(n,k). Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000045(n+1), A006131(n), A015445(n), A168579(n), A122999(n) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Dec 02 2009
T(n,k) = binomial((n+k)/2,k) if (n+k) is even; otherwise T(n,k)=0.
G.f.: (1-z^2)/(1-t*z-z^2) if offset is 1.
T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = 1, T(0,1) = 0. - Philippe Deléham, Feb 09 2012
Sum_{k=0..n} T(n,k)^2 = A051286(n). - Philippe Deléham, Feb 09 2012
From R. J. Mathar, Feb 04 2022: (Start)
Sum_{k=0..n} T(n,k)*k = A001629(n+1).
EXAMPLE
The triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
0: 1
1: 0 1
2: 1 0 1
3: 0 2 0 1
4: 1 0 3 0 1
5: 0 3 0 4 0 1
6: 1 0 6 0 5 0 1
7: 0 4 0 10 0 6 0 1
8: 1 0 10 0 15 0 7 0 1
9: 0 5 0 20 0 21 0 8 0 1
10: 1 0 15 0 35 0 28 0 9 0 1
11: 0 6 0 35 0 56 0 36 0 10 0 1
12: 1 0 21 0 70 0 84 0 45 0 11 0 1
13: 0 7 0 56 0 126 0 120 0 55 0 12 0 1
14: 1 0 28 0 126 0 210 0 165 0 66 0 13 0 1
15: 0 8 0 84 0 252 0 330 0 220 0 78 0 14 0 1
... reformatted by Wolfdieter Lang, Jul 29 2014.
------------------------------------------------------------------------
MAPLE
MATHEMATICA
Table[If[EvenQ[n + k], Binomial[(n + k)/2, k], 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 16 2017 *)
PROG
(PARI) T(n, k) = if ((n+k) % 2, 0, binomial((n+k)/2, k));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print(); ); \\ Michel Marcus, Oct 09 2016
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Nov 29 2009
EXTENSIONS
Typo in name corrected (1(1-x^2) changed to 1/(1-x^2)) by Wolfdieter Lang, Nov 20 2010
STATUS
approved