cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A202480 Riordan array (1/(1-x), x(2x-1)/(1-x)^2).

Original entry on oeis.org

1, 1, -1, 1, -1, 1, 1, 0, 1, -1, 1, 2, -1, -1, 1, 1, 5, -5, 2, 1, -1, 1, 9, -10, 8, -3, -1, 1, 1, 14, -14, 14, -11, 4, 1, -1, 1, 20, -14, 14, -17, 14, -5, -1, 1, 1, 27, -6, 0, -9, 19, -17, 6, 1, -1
Offset: 0

Views

Author

Philippe Deléham, Dec 20 2011

Keywords

Comments

Row sums are Fibonacci(n-1) = A000045(n-1).
Diagonal sums are A078003(n).
(Sum_{j, 0<=j<=k} T(k,j))/(1-2x)^k gives g.f. of column A165241(n+k-1,k-1) in triangular array in A165241.

Examples

			Triangle begins :
1
1, -1
1, -1, 1
1, 0, 1, -1
1, 2, -1, -1, 1
1, 5, -5, 2, 1, -1
1, 9, -10, 8, -3, -1, 1
1, 14, -14, 14, -11, 4, 1, -1
(1+x^2-x^3)/(1-2x)^3 is the g.f of column A165241(n+2,2) := 1, 6, 25, 85, 258, 728, 1952, 5040, ...

Crossrefs

Cf. A000012, A080956, A000045, A078003, A124341, A165241

Formula

T(n,k) = 2*T(n-1,k) + 2*T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).
T(n,k) = (-1)^n*A124341(n,k).

A124342 Expansion of (1+x)/(1+2x-2x^3).

Original entry on oeis.org

1, -1, 2, -2, 2, 0, -4, 12, -24, 40, -56, 64, -48, -16, 160, -416, 800, -1280, 1728, -1856, 1152, 1152, -6016, 14336, -26368, 40704, -52736, 52736, -24064, -57344, 220160, -488448, 862208, -1284096, 1591296, -1458176, 348160
Offset: 0

Views

Author

Paul Barry, Oct 26 2006

Keywords

Comments

Diagonal sums of A124341. Binomial transform has g.f. (1-x)/(1-x-x^2-x^3).

Links

Formula

a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, (-1)^(n-k-j)*C(n-k,j)*C(k,j-k)}}
a(n) = (-1)^(n+1)*A073358(n+1). - R. J. Mathar, Feb 04 2014
a(n) = A077988(n-1)+A077988(n). - R. J. Mathar, Jan 25 2016

A146314 Inverse of Riordan array A127543.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -2, -1, 1, 1, 1, -5, -5, -2, 1, 1, 1, -9, -10, -8, -3, 1, 1, 1, -14, -14, -14, -11, -4, 1, 1, 1, -20, -14, -14, -17, -14, -5, 1, 1, 1, -27, -6, 0, -9, -19, -17, -6, 1, 1
Offset: 0

Views

Author

Philippe Deléham, Oct 29 2008

Keywords

Comments

A007318*A109466 as infinite lower triangular matrices.
Riordan array (1/(1-x), x*(1-2*x)/(1-x)^2). - Philippe Deléham, Jan 27 2014

Examples

			Triangle begins :
1;
1, 1;
1, 1, 1;
1, 0, 1, 1;
1, -2, -1, 1, 1;
1, -5, -5, -2, 1, 1;
1, -9, -10, -8, -3, 1, 1;
1, -14, -14, -14, -11, -4, 1, 1;
1, -20, -14, -14, -17, -14, -5, 1, 1;
1, -27, -6, 0, -9, -19, -17, -6, 1, 1;...

Crossrefs

Cf. A124341

Formula

Number triangle T(n,k)=Sum_{j=0..n, (-1)^(n-j)*C(k,n-j)*C(n,j-k)}.
Showing 1-3 of 3 results.

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