%I #70 Oct 09 2024 09:11:23

%S 1,0,1,1,0,1,0,2,0,1,1,0,3,0,1,0,3,0,4,0,1,1,0,6,0,5,0,1,0,4,0,10,0,6,

%T 0,1,1,0,10,0,15,0,7,0,1,0,5,0,20,0,21,0,8,0,1,1,0,15,0,35,0,28,0,9,0,

%U 1,0,6,0,35,0,56,0,36,0,10,0,1,1,0,21,0,70,0,84,0,45,0,11,0,1

%N Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310.

%C Row sums: A000045(n+1), Fibonacci numbers.

%C A168561*A007318 = A037027, as lower triangular matrices. Diagonal sums : A077957. - _Philippe Deléham_, Dec 02 2009

%C 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.

%C 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

%H G. C. Greubel, <a href="/A168561/b168561.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H J.P. Allouche and M. Mendès-France, <a href="http://arxiv.org/abs/1202.0211">Stern-Brocot polynomials and power series</a>, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From _N. J. A. Sloane_, May 10 2012

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.

%F 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

%F 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

%F T(n,k) = binomial((n+k)/2,k) if (n+k) is even; otherwise T(n,k)=0.

%F G.f.: (1-z^2)/(1-t*z-z^2) if offset is 1.

%F 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

%F Sum_{k=0..n} T(n,k)^2 = A051286(n). - _Philippe Deléham_, Feb 09 2012

%F From _R. J. Mathar_, Feb 04 2022: (Start)

%F Sum_{k=0..n} T(n,k)*k = A001629(n+1).

%F Sum_{k=0..n} T(n,k)*k^2 = 0,1,4,11,... = 2*A055243(n)-A099920(n+1).

%F Sum_{k=0..n} T(n,k)*k^3 = 0,1,8,29,88,236,... = 12*A055243(n) -6*A001629(n+2) +A001629(n+1)-6*(A001872(n)-2*A001872(n-1)). (End)

%e The triangle T(n,k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

%e 0: 1

%e 1: 0 1

%e 2: 1 0 1

%e 3: 0 2 0 1

%e 4: 1 0 3 0 1

%e 5: 0 3 0 4 0 1

%e 6: 1 0 6 0 5 0 1

%e 7: 0 4 0 10 0 6 0 1

%e 8: 1 0 10 0 15 0 7 0 1

%e 9: 0 5 0 20 0 21 0 8 0 1

%e 10: 1 0 15 0 35 0 28 0 9 0 1

%e 11: 0 6 0 35 0 56 0 36 0 10 0 1

%e 12: 1 0 21 0 70 0 84 0 45 0 11 0 1

%e 13: 0 7 0 56 0 126 0 120 0 55 0 12 0 1

%e 14: 1 0 28 0 126 0 210 0 165 0 66 0 13 0 1

%e 15: 0 8 0 84 0 252 0 330 0 220 0 78 0 14 0 1

%e ... reformatted by _Wolfdieter Lang_, Jul 29 2014.

%e ------------------------------------------------------------------------

%p A168561:=proc(n,k) if n-k mod 2 = 0 then binomial((n+k)/2,k) else 0 fi end proc:

%p seq(seq(A168561(n,k),k=0..n),n=0..12) ; # yields sequence in triangular form

%t Table[If[EvenQ[n + k], Binomial[(n + k)/2, k], 0], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Apr 16 2017 *)

%o (PARI) T(n,k) = if ((n+k) % 2, 0, binomial((n+k)/2,k));

%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ _Michel Marcus_, Oct 09 2016

%Y Cf. A162515 (rows reversed), A112552, A102426 (deflated).

%K easy,nonn,tabl

%O 0,8

%A _Philippe Deléham_, Nov 29 2009

%E Typo in name corrected (1(1-x^2) changed to 1/(1-x^2)) by _Wolfdieter Lang_, Nov 20 2010