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 A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310. 17
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 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-tz-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 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 T:=proc(n, k) if n-k mod 2 = 0 then binomial((n+k)/2, k) else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form 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 Cf. A162515 (rows reversed), A112552. Sequence in context: A180649 A191238 A049310 * A253190 A293307 A293293 Adjacent sequences:  A168558 A168559 A168560 * A168562 A168563 A168564 KEYWORD easy,nonn,tabl 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

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Last modified July 23 22:13 EDT 2021. Contains 346265 sequences. (Running on oeis4.)