The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A038137 Reflection of A037027: T(n,m) = U(n,n-m), m=0..n, where U is as in A037027. 9
1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 8, 1, 6, 20, 40, 51, 38, 13, 1, 7, 27, 65, 105, 111, 71, 21, 1, 8, 35, 98, 190, 256, 233, 130, 34, 1, 9, 44, 140, 315, 511, 594, 474, 235, 55, 1, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 1, 11, 65, 255 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Number of lattice paths from (0,0) to (n,k) using steps (1,0), (1,1), (2,2). - Joerg Arndt, Jul 01 2011
The n-th diagonal D(n) = {T(n,0), T(n+1,1), ..., T(n+m,m), ...} of the triangle has generating function F(x) = 1/(1 - x - x^2)^(n+1) for n = 0,1,2,.... - L. Edson Jeffery, Mar 20 2011
Let p(n,x) denote the Fibonacci polynomial, defined by p(1,x) = 1, p(2,x) = x, and p(n,x) = x*p(n-1,x) + p(n-2,x) for n >= 3. Let q(n,x) be the numerator polynomial of the rational function p(n, 1 + 1/x). The coefficients of the polynomial q(n,x) are given by the (n-1)-th row of T(n,k). E.g., p(5,x) = 1 + 3*x^2 + x^4 gives q(5,x) = 1 + 4*x + 9*x^2 + 10*x^2 + 5*x^4. - Clark Kimberling, Nov 04 2013
LINKS
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Pieter Moree, Convoluted convolved Fibonacci numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
Eric Weisstein's World of Mathematics, Fibonacci polynomial.
FORMULA
From Paul Barry, Oct 24 2005: (Start)
G.f.: 1/(1 - x - x*y - x^2*y^2).
T(n,k) = Sum_{j=0..n} C((n+j)/2, j) * (1 + (-1)^(n+j)) * C(j, n-k)/2. (End)
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(n,k) = 0 if n < 0 or if n < k, and T(0,0) = 1. - Philippe Deléham, Nov 30 2006
Sum_{k=0..n} (-1)^k*T(n,k) = A059841(n). - Philippe Deléham, Nov 30 2006
T(n,k) = A208336(n+1,k).- Philippe Deléham, Apr 05 2012
EXAMPLE
Triangle T(n,k) (with rows n >= 0 and columns 0 <= k <= n) begins
1;
1, 1;
1, 2, 2;
1, 3, 5, 3;
1, 4, 9, 10, 5;
1, 5, 14, 22, 20, 8;
1, 6, 20, 40, 51, 38, 13;
1, 7, 27, 65, 105, 111, 71, 21;
...
PROG
(Haskell)
a038137 n k = a038137_tabl !! n !! k
a038137_row n = a038137_tabl !! n
a038137_tabl = map reverse a037027_tabl
-- Reinhard Zumkeller, Jul 08 2012
CROSSREFS
Row sums are Pell numbers A000129.
Diagonal sums are unsigned version of A077930.
Sequence in context: A140767 A060850 A208336 * A073133 A106179 A081572
KEYWORD
easy,nonn,tabl
AUTHOR
EXTENSIONS
Title corrected by L. Edson Jeffery, Apr 23 2011
Corrected by Philippe Deléham, Apr 05 2012
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 22 20:45 EDT 2024. Contains 372758 sequences. (Running on oeis4.)