A104041 -id:A104041 - cp's OEIS frontend

A140685 Triangle T(n,k) read by rows: T(n,k) = 1 if n is odd and k=(n-1)/2; T(n,k) = 2 otherwise.

1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Roger L. Bagula and Mats Granvik, Jul 11 2008

Row sums give A042948.

			The triangle starts in row n=1 with columns 0 <= k < n as:
  1;
  2, 2;
  2, 1, 2;
  2, 2, 2, 2;
  2, 2, 1, 2, 2;
  2, 2, 2, 2, 2, 2;
  2, 2, 2, 1, 2, 2, 2;
  2, 2, 2, 2, 2, 2, 2, 2;
  2, 2, 2, 2, 1, 2, 2, 2, 2;
  2, 2, 2, 2, 2, 2, 2, 2, 2, 2;

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 rows of the triangle

Cf. A002024, A002260, A104041.

(define (A140685 n) (A140685tr (A002024 n) (- (A002260 n) 1)))
(define (A140685tr n k) (if (and (odd? n) (= k (/ (- n 1) 2))) 1 2))

Definition simplified by the Assoc. Eds. of the OEIS, Oct 12 2010

More terms from Antti Karttunen, Oct 10 2017

A104040 Triangular matrix T, read by rows, such that row n equals the absolute values of column (n+1) in the matrix inverse T^-1 (with extrapolated zeros): T(n,k) = -Sum_{j=1..[n+1/2]} (-1)^jT(n-j+1,n-2j+1)*T(n-j,k) with T(n,n)=1 (n>=0) and T(n,n-1)=n (n>=1).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 8, 8, 8, 4, 1, 16, 16, 20, 12, 5, 1, 32, 32, 48, 32, 18, 6, 1, 64, 64, 112, 80, 56, 24, 7, 1, 128, 128, 256, 192, 160, 80, 32, 8, 1, 256, 256, 576, 448, 432, 240, 120, 40, 9, 1, 512, 512, 1280, 1024, 1120, 672, 400, 160, 50, 10, 1, 1024, 1024, 2816

Offset: 0

Paul D. Hanna, Mar 02 2005

Row sums are the Pell numbers A000129. Let A(x,y) be the g.f. of T and B(x,y) be the g.f. of T^-1; then B(x,y)=(A(-x^2*y,-1/x)-1)/(x*y) and A(x,y)=1+x*y*B(-1/y,-x*y^2).

			Rows of T begin:
1;
1,1;
2,2,1;
4,4,3,1;
8,8,8,4,1;
16,16,20,12,5,1;
32,32,48,32,18,6,1;
64,64,112,80,56,24,7,1;
128,128,256,192,160,80,32,8,1; ...
The matrix inverse T^-1 equals triangle A104041:
1;
-1,1;
0,-2,1;
0,2,-3,1;
0,0,4,-4,1;
0,0,-4,8,-5,1;
0,0,0,-8,12,-6,1;
0,0,0,8,-20,18,-7,1; ...
the columns of T^-1 equal rows of T in absolute value.

Cf. A104041, A000129.

PARI
```
T(n,k)=if(n
				
```
PARI
```
T(n,k)=if(n1 && k>1,T(n-2,k-2)))))
```

T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X*Y)/(1-2*X-X^2*Y^2),n,x),k,y)

G.f.: A(x, y) = (1-x+x*y)/(1-2*x-x^2*y^2). T(n, k) = 2*T(n-1, k) + T(n-2, k-2) (n>=k>=2) with T(0, 0)=T(1, 0)=T(1, 1)=1.

cp's OEIS Frontend

A140685 Triangle T(n,k) read by rows: T(n,k) = 1 if n is odd and k=(n-1)/2; T(n,k) = 2 otherwise.

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A104040 Triangular matrix T, read by rows, such that row n equals the absolute values of column (n+1) in the matrix inverse T^-1 (with extrapolated zeros): T(n,k) = -Sum_{j=1..[n+1/2]} (-1)^jT(n-j+1,n-2j+1)*T(n-j,k) with T(n,n)=1 (n>=0) and T(n,n-1)=n (n>=1).

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cp's OEIS Frontend

A140685 Triangle T(n,k) read by rows: T(n,k) = 1 if n is odd and k=(n-1)/2; T(n,k) = 2 otherwise.

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Author

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Scheme

Extensions

A104040 Triangular matrix T, read by rows, such that row n equals the absolute values of column (n+1) in the matrix inverse T^-1 (with extrapolated zeros): T(n,k) = -Sum_{j=1..[n+1/2]} (-1)^j*T(n-j+1,n-2*j+1)*T(n-j,k) with T(n,n)=1 (n>=0) and T(n,n-1)=n (n>=1).

Views

Author

Keywords

Comments

Examples

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Programs

PARI

PARI

PARI

Formula

A104040 Triangular matrix T, read by rows, such that row n equals the absolute values of column (n+1) in the matrix inverse T^-1 (with extrapolated zeros): T(n,k) = -Sum_{j=1..[n+1/2]} (-1)^jT(n-j+1,n-2j+1)*T(n-j,k) with T(n,n)=1 (n>=0) and T(n,n-1)=n (n>=1).