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.

A117185 Riordan array ((1-x^2)/(1+x+2x^2+x^3+x^4),x/(1+x^2)).

Original entry on oeis.org

1, -1, 1, -2, -1, 1, 3, -3, -1, 1, 1, 4, -4, -1, 1, -4, 4, 5, -5, -1, 1, 1, -8, 8, 6, -6, -1, 1, 3, -3, -13, 13, 7, -7, -1, 1, -2, 11, -11, -19, 19, 8, -8, -1, 1, -1, 1, 24, -24, -26, 26, 9, -9, -1, 1, 1, -12, 12, 43, -43, -34, 34, 10, -10, -1, 1
Offset: 0

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Author

Paul Barry, Mar 01 2006

Keywords

Comments

Row sums are A117188. Diagonal sums are A099837. Inverse of A117184.

Examples

			Triangle begins
1,
-1, 1,
-2, -1, 1,
3, -3, -1, 1,
1, 4, -4, -1, 1,
-4, 4, 5, -5, -1, 1,
1, -8, 8, 6, -6, -1, 1,
3, -3, -13, 13, 7, -7, -1, 1

A117186 Expansion of (1+x)c(x^2)/((1-xc(x^2))*sqrt(1-4x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 5, 9, 21, 38, 86, 157, 349, 642, 1410, 2610, 5682, 10572, 22860, 42717, 91869, 172298, 368906, 694054, 1480486, 2793012, 5938740, 11230834, 23813746, 45131348, 95462996, 181268292, 382594884, 727747608, 1533053976
Offset: 0

Views

Author

Paul Barry, Mar 01 2006

Keywords

Comments

Row sums of triangle A117184.

Formula

G.f.: (1+x)(sqrt(1-4x^2)+2x-1)/(2x(1-2x)*sqrt(1-4x^2)); a(n)=sum{k=0..n, C(n+1,(n+k)/2+1)(1+(-1)^(n-k))/2+C(n,(n+k)/2+1/2)(1-(-1)^(n-k))/2}.
G.f.: (1+x)(1+2x-sqrt(1-4x^2))/(2x(1-4x^2)); a(n)=(3*2^n-binomial(2*floor((n+1)/2),floor((n+1)/2)))/2; - Paul Barry, Jan 20 2008
Conjecture: a(n) = A058622(n) + A058622(n+1). [R. J. Mathar, Nov 21 2008]
Conjecture: -(n+1)*a(n) +(n+1)*a(n-1) +2*(3*n-2)*a(n-2) -4*n*a(n-3) +8*(3-n)*a(n-4)=0. - R. J. Mathar, Nov 15 2011

A117187 Expansion of (1+x)c(x^2)/((1-x^2*c(x^2))sqrt(1-4x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 4, 4, 15, 15, 56, 56, 210, 210, 792, 792, 3003, 3003, 11440, 11440, 43758, 43758, 167960, 167960, 646646, 646646, 2496144, 2496144, 9657700, 9657700, 37442160, 37442160, 145422675, 145422675, 565722720
Offset: 0

Views

Author

Paul Barry, Mar 01 2006

Keywords

Comments

Diagonal sums of triangle A117184. Also second column of triangle A117184.

Formula

E.g.f.: dif(Bessel_I(1,2x)+Bessel_I(2,2x)+Bessel_I(3,2x),x);
a(n)=sum{k=0..n, C(n-k+1,n/2+1)(1+(-1)^n)/2+C(n-k, n/2+1/2)(1-(-1)^n)/2}.
Conjecture D-finite with recurrence: (n+4)*(31n-12)*a(n) +(37n^2+24n-156)*a(n-1) -4*(31n^2+81n+24)*a(n-2) -4*(37n+24)*(n-1)*a(n-3) =0. - R. J. Mathar, Dec 10 2011
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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