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A168561 Riordan array (1/(1-x^2), x/(1-x^2)). Unsigned version of A049310. 13
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. Mendes France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012. - From N. J. A. Sloane, May 10 2012

T. Copeland, Addendum to Elliptic Lie Triad

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

FORMULA

Sum_{k, 0<=k<=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<=k<=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; else 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<=k<=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. A112552.

Sequence in context: A180649 A191238 A049310 * A253190 A293307 A293293

Adjacent sequences:  A168558 A168559 A168560 * A168562 A168563 A168564

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Nov 29 2009

EXTENSIONS

Corrected misprint in name: 1(1-x^2)->1/(1-x^2)

STATUS

approved

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Last modified January 16 15:53 EST 2019. Contains 319195 sequences. (Running on oeis4.)