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.

A171567 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A168491.

Original entry on oeis.org

1, -1, 1, 2, -2, 1, -5, 5, -3, 1, 14, -14, 9, -4, 1, -42, 42, -28, 14, -5, 1, 132, -132, 90, -48, 20, -6, 1, -429, 429, -297, 165, -75, 27, -7, 1, 1430, -1430, 1001, -572, 275, -110, 35, -8, 1, -4862, 4862, -3432, 2002, -1001, 429, -154, 44, -9, 1
Offset: 0

Views

Author

Philippe Deléham, Dec 11 2009

Keywords

Comments

Equal to B^(-2)*A054336, B = A007318. T(n,0)=(-1)^n*A000108(n). Unsigned version in A033184.

Examples

			Triangle begins : 1 ; -1,1 ; 2,-2,1 ; -5,5,-3,1 ; 14,-14,9,-4,1 ;...

Crossrefs

Cf. A033184, A171488, A054336, A154930

Formula

T(n,k) = (-1)^(n-k)*(k+1)*binomial(2n-k, n-k)/(n+1), 0<=k<=n; else 0.
T(n,k) = T(n-1,k-1) - T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012

A154930 Inverse of Fibonacci convolution array A154929.

Original entry on oeis.org

1, -2, 1, 5, -4, 1, -15, 14, -6, 1, 51, -50, 27, -8, 1, -188, 187, -113, 44, -10, 1, 731, -730, 468, -212, 65, -12, 1, -2950, 2949, -1956, 970, -355, 90, -14, 1, 12235, -12234, 8291, -4356, 1785, -550, 119, -16, 1, -51822, 51821, -35643, 19474, -8612, 3021
Offset: 0

Views

Author

Paul Barry, Jan 17 2009

Keywords

Comments

Alternating sign version of A104259. Row sums are (-1)^n*A033321. First column is (-1)^n*A007317.

Examples

			Triangle begins
1,
-2, 1,
5, -4, 1,
-15, 14, -6, 1,
51, -50, 27, -8, 1,
-188, 187, -113, 44, -10, 1,
731, -730, 468, -212, 65, -12, 1,
-2950, 2949, -1956, 970, -355, 90, -14, 1
Production array is
-2, 1,
1, -2, 1,
-1, 1, -2, 1,
1, -1, 1, -2, 1,
-1, 1, -1, 1, -2, 1,
1, -1, 1, -1, 1, -2, 1,
-1, 1, -1, 1, -1, 1, -2, 1
or ((1-x-x^2)/(1+x),x) beheaded.

Crossrefs

Cf. A171567, A054336, A171488, A035324.

Formula

Riordan array ((1/(1+x))c(-x/(1+x)), (x/(1+x))c(x/(1+x))), c(x) the g.f. of A000108;
Riordan array ((sqrt(1+6x+5x^2)-x-1)/(2x(1+x)),(sqrt(1+6x+5x^2)-x-1)/ (2(1+x)));
Triangle T(n,k) = sum{j=0..n, (-1)^(n-k)*C(n,j)*C(2j-k,j-k)(k+1)/(j+1)}.
T(n,k) = T(n-1,k-1) -2*T(n-1,k) + Sum_{i, i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012

A171505 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A059738.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 34, 29, 9, 1, 117, 128, 57, 12, 1, 405, 538, 309, 94, 15, 1, 1407, 2192, 1533, 604, 140, 18, 1, 4899, 8740, 7179, 3453, 1040, 195, 21, 1, 17083, 34296, 32278, 18264, 6730, 1644, 259, 24, 1, 59629, 132929, 140790, 91372, 39668, 11877, 2443
Offset: 0

Views

Author

Philippe Deléham, Dec 10 2009

Keywords

Comments

Equal to B*A096164 = A171488*B, B=A007318.

Examples

			Triangle begins :
1 ;
3, 1 ;
10, 6, 1 ;
34, 29, 9, 1 ;
117, 128, 57, 12, 1 ; ...

Crossrefs

Cf. A097609, A064189, A171488, A096164

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -3, -2, -1, 0 respectively.
T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*(-2)^i. - Philippe Deléham, Feb 23 2012
Showing 1-3 of 3 results.

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