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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), 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, p(n,x) = x*p(n-1,x) + p(n-2,x).  Let q(n,x) be the numerator polynomial of the rational function p(n, 1 + 1/x).  The coefficients of the polynomials q(n,x)  are given by A038137; 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

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened

P. Moree, Convoluted convolved Fibonacci numbers

FORMULA

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}. - Paul Barry, Oct 24 2005

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, T(0,0)=1. - Philippe Deléham, Nov 30 2006

Sum_{k, 0<=k<=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 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;

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

Adjacent sequences:  A038134 A038135 A038136 * A038138 A038139 A038140

KEYWORD

easy,nonn,tabl

AUTHOR

Floor van Lamoen

EXTENSIONS

Title corrected by L. Edson Jeffery, Apr 23 2011

Corrected by Philippe Deléham, Apr 05 2012

STATUS

approved

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Last modified June 24 15:07 EDT 2017. Contains 288697 sequences.