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 1-4 of 4 results.

A111959 Renewal array for aerated central binomial coefficients.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 6, 0, 6, 0, 1, 0, 16, 0, 8, 0, 1, 20, 0, 30, 0, 10, 0, 1, 0, 64, 0, 48, 0, 12, 0, 1, 70, 0, 140, 0, 70, 0, 14, 0, 1, 0, 256, 0, 256, 0, 96, 0, 16, 0, 1, 252, 0, 630, 0, 420, 0, 126, 0, 18, 0, 1, 0, 1024, 0, 1280, 0, 640, 0, 160, 0, 20, 0, 1, 924, 0, 2772, 0
Offset: 0

Views

Author

Paul Barry, Aug 23 2005

Keywords

Comments

Row sums are A098615.
Binomial transform (product with C(n,k)) is A111960.
Diagonal sums are A026671 (with interpolated zeros).
Inverse is (1/sqrt(1+4x^2),x/sqrt(1+4x^2)), or (sqrt(-1))^(n-k)*T(n,k). [corrected by Peter Bala, Aug 13 2021]
The Riordan array (1,x/sqrt(1-4*x^2)) is the same array with an additional column of zeros (besides the top element 1) added to the left. - Vladimir Kruchinin, Feb 17 2011

Examples

			From _Peter Bala_, Aug 13 2021: (Start)
Triangle begins
  1;
  0,  1;
  2,  0, 1;
  0,  4, 0, 1;
  6,  0, 6, 0, 1;
  0, 16, 0, 8, 0, 1;
Infinitesimal generator begins
  0;
  0, 0;
  2, 0, 0;
  0, 4, 0, 0;
  0, 0, 6, 0, 0;
  0, 0, 0, 8, 0, 0; (End)

Links

Crossrefs

Cf. A054335, A038231, A046521.

Formula

Riordan array (1/sqrt(1-4x^2), x/sqrt(1-4x^2)); number triangle T(n, k)=(1+(-1)^(n-k))*binomial((n-1)/2, (n-k)/2)*2^(n-k)/2.
G.f.: 1/(1-xy-2x^2/(1-x^2/(1-x^2/(1-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
From Peter Bala, Aug 13 2021: (Start)
T(2*n,2*k) = A046521(n,k); T(2*n+1,2*k+1) = A038231(n,k).
The row entries, read from right to left, are the coefficients in the n-th order Taylor polynomial of (sqrt(1 + 4*x^2))^((n-1)/2) at x = 0.
The infinitesimal generator of this array has the sequence [2, 4, 6, 8, 10, ...] on the second subdiagonal below the main diagonal and zeros elsewhere.
The m-th power of the array is the Riordan array (1/sqrt(1 - 4*m*x^2), x/sqrt(1 - 4*m*x^2)) with entries given by sqrt(m)^(n-k)*T(n,k). (End)

A111961 Expansion of 1/(sqrt(1-2x-3x^2)-x).

Original entry on oeis.org

1, 2, 6, 18, 56, 176, 558, 1778, 5686, 18230, 58558, 188366, 606588, 1955044, 6305418, 20347342, 65689088, 212146400, 685342218, 2214556478, 7157409064, 23136645472, 74801223162, 241863933094, 782131232390, 2529458676326
Offset: 0

Views

Author

Paul Barry, Aug 23 2005

Keywords

Comments

Row sums of A111960.
A transform of the Fibonacci numbers. - Paul Barry, Sep 23 2005
Apparently the Motzkin transform of (0 followed by A128588). - R. J. Mathar, Dec 11 2008
Inverse binomial transform of A026671. - Philippe Deléham, Feb 11 2009
Hankel transform is 2^n. - Paul Barry, Mar 02 2010

Links

Crossrefs

Cf. A001006, A026671, A111960, A128588.

Programs

  • Mathematica
    CoefficientList[Series[1/(Sqrt[1-2*x-3*x^2]-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C((j-1)/2, (j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2.
a(n) = Sum_{k=0..n} F(k+1)*Sum_{i=0..floor((n-k)/2)} C(n, i)*C(n-i, i+k)/(i+k+1). - Paul Barry, Sep 23 2005
G.f.: M(x)^2/(2*M(x)-M(x)^2), where M(x) is the g.f. of the Motzkin numbers A001006. - Paul Barry, Feb 03 2006
G.f.: 1/(1-2x/(1-x/(1-x^2/(1-x/(1-x/91-x^2/(1-x/(1-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Mar 02 2010
D-finite with recurrence: n*a(n) + (-4*n+3)*a(n-1) + 3*(-n+1)*a(n-2) + 2*(7*n-15)*a(n-3) + 12*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2012
a(n) ~ (1+sqrt(5))^n / sqrt(5). - Vaclav Kotesovec, Feb 08 2014

A111963 Inverse of renewal array for central trinomial numbers.

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 3, -1, -3, 1, 1, 8, 0, -4, 1, -9, -3, 14, 2, -5, 1, 1, -26, -15, 20, 5, -6, 1, 27, 27, -45, -37, 25, 9, -7, 1, -13, 76, 98, -56, -70, 28, 14, -8, 1, -81, -135, 108, 228, -46, -114, 28, 20, -9, 1, 67, -202, -459, 48, 420, 0, -168, 24, 27, -10, 1, 243, 567, -135, -1035, -210, 662, 98, -230, 15, 35, -11, 1, -285
Offset: 0

Views

Author

Paul Barry, Aug 23 2005

Keywords

Comments

Row sums have g.f. 1/sqrt(1+4x^2) [alternating sign central binomial numbers with interpolated zeros]. Diagonal sums are A111964. Inverse of A111960. Factors as (1/sqrt(1+4x^2),x/sqrt(1+4x^2))*(1/(1+x),x/(1+x)).

Examples

			Triangle begins
1;
-1,1;
-1,-2,1;
3,-1,-3,1;
1,8,0,-4,1;
-9,-3,14,2,-5,1;
1,-26,-15,20,5,-6,1;

Formula

Riordan array (1/(sqrt(1+4x^2)+x), x/(sqrt(1+4x^2)+x)); Number triangle T(n, k)=sum{i=0..floor(n/2), C(2i+k-n-1, k)*C((2i-n-1)/2, i)(-1)^n*4^i}.

A111962 Expansion of 1/(sqrt(1-2x-3x^2)-x^2).

Original entry on oeis.org

1, 1, 4, 9, 27, 74, 215, 619, 1808, 5293, 15579, 45986, 136141, 403937, 1200854, 3575835, 10663105, 31836508, 95156251, 284684303, 852427700, 2554346249, 7659427811, 22981483554, 68992151795, 207223579671, 622697586944
Offset: 0

Views

Author

Paul Barry, Aug 23 2005

Keywords

Comments

Diagonal sums of A111960.

Formula

a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, C(n-k, j)*C((j-1)/2, (j-k)/2)*2^(j-k)*(1+(-1)^(j-k))/2}}
D-finite with recurrence: n*a(n) +(-4*n+3)*a(n-1) +(-2*n+3)*a(n-2) +3*(4*n-9)*a(n-3) +(8*n-27)*a(n-4) +(2*n-3)*a(n-5) +3*(n-3)*a(n-6)=0. - R. J. Mathar, Jan 24 2020
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.