cp's OEIS Frontend

Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.

From Andrew V. Sutherland, Feb 29 2008: (Start)

Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.

Moment sequence of the trace of a random matrix in G = SO(2). If X = tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n].

Also the moment sequence of the trace of the k-th power of a random matrix in USp(2) = SU(2), for all k > 2.

(End)

From Paul Barry, Aug 10 2009: (Start)

The Hankel transform of 0,1,0,2,0,6,... is 0,-1,0,4,0,-16,0,... with general term I*(-4)^(n/2)(1 - (-1)^n)/4, I = sqrt(-1).

The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1 + x/sqrt(1 - 4*x^2)) is A164111. (End)

a(n) = A204293(2*n,n): central terms of the triangle in A204293. - Reinhard Zumkeller, Jan 14 2012

a(n) is the total number of closed walks (round trips) of length n on the graph P_N (a line with N nodes and N-1 edges), divided by N, in the limit N -> infinity. See a comment on A198632 and a link under A201198. - Wolfdieter Lang, Oct 10 2012

a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=0; also a(n)=T(2n,n).

Partial sums of A085362. Number of bilateral Schroeder paths (i.e., lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at odd (positive or negative) levels. Example: a(2)=11 because we have HUD, UDH, UDUD, UUDD, UDDU, their reflections in the x-axis and HH. - Emeric Deutsch, Jan 30 2004

Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733. - Philippe Deléham, Oct 02 2007

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in three colors. - N-E. Fahssi, Feb 05 2008

Equals INVERT transform of A109033: (1, 2, 6, 22, 88, ...), INVERTi transform of A111966, binomial transform of A000984, and inverse Binomial transform of A081671. Convolved with A002212: (1, 3, 10, 36, ...) = A026376: (1, 6, 30, 144, ...). Equals convolution square root of A003463: (1, 6, 31, 156, 781, 3906, ...). - Gary W. Adamson, May 17 2009

Diagonal of array with rational generating function 1/(1 - (x^2 + 3*x*y + y^2)). - Gheorghe Coserea, Jul 29 2018

a(n) == 0 (mod 3) if and only if n is in A081606. - Fabio Visonà, Aug 03 2023

Column 1 yields A064613. Row sums yield A081671.

Triangle T(n,k), 0 <= k <= n, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k+1). - Philippe Deléham, Feb 27 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

6^n = ((n+1)-th row terms) dot (first n+1 odd integers). Example: 6^4 = 1296 = (150, 204, 84, 15, 1) dot (1, 3, 5, 7, 9) = (150 + 612 + 420 + 105 + 9)= 1296. - Gary W. Adamson, Jun 15 2011

From Peter Bala, Sep 06 2022: (Start)

The following assume the row and column indexing start at 0.

Riordan array (f(x), x*g(x)), where f(x) = (1 - sqrt((1 - 6*x)/(1 - 2*x)))/(2*x) is the o.g.f. of A064613 and g(x) = (1 - 4*x - sqrt(1 - 8*x + 12*x^2))/(2*x^2) is the o.g.f. of A005572.

The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x)*(1 + 4*x + x^2)^n expanded about the point x = 0.

T(n,k) = a(n,k) - a(n,k+1), where a(n,k) = Sum_{j = 0..n} binomial(n,j)* binomial(j,n-k-j)*4^(2*j+k-n). (End)

Binomial transform of A081671. 3rd binomial transform of A000984. Binomial transform is A098410.

Largest coefficient of (1+5*x+x^2)^n; row sums of triangle in A126331. - Philippe Deléham, Oct 02 2007

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in five colors. - N-E. Fahssi, Feb 05 2008

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have five colors. - N-E. Fahssi, Mar 31 2008

Diagonal of rational function 1/(1 - (x^2 + 5*x*y + y^2)). - Gheorghe Coserea, Aug 01 2018

Number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g., a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD and UHUDD. - Emeric Deutsch, Dec 28 2003

Number of left steps in all skew Dyck paths of semilength n+1. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL and five Dyck paths that have no left steps) we have altogether 6 left steps. - Emeric Deutsch, Aug 05 2007

From Gary W. Adamson, May 17 2009: (Start)

Equals A026378 (1, 4, 17, 75, ...) convolved with A007317 (1, 2, 5, 15, 51, ...).

Equals A081671 (1, 3, 11, 45, ...) convolved with A002212 (1, 3, 10, 36, 137, ...).

(End)

Fifth binomial transform of A000984. In general, the k-th binomial transform of A000984 will have g.f. 1/(sqrt(1-k*x)*sqrt(1-(k+4)*x)) and a(n)=sum{i=0..n, C(n,i)C(2i,i)k^(n-i)}.

Diagonal of rational function 1/(1 - (x^2 + 7*x*y + y^2)). - Gheorghe Coserea, Aug 03 2018

A(n,k) is the k-th binomial transform of A126869 evaluated at n.

A Grand Motzkin path is a path in the half-plane x >= 0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).

From Peter Bala, Feb 11 2017: (Start)

Consider an infinite 1-dimensional integer lattice with an oriented self-loop at each vertex. Then T(n,k) equals the number of walks of length n from a vertex to itself having k loops. There is a bijection between such walks and Grand Motzkin paths which takes a right step and a left step on the lattice to an up step U and a down step D of a Grand Motzkin path respectively, and takes traversing a loop on the lattice to the horizontal step H. See A282252 for the corresponding triangle of walks on a 2-dimensional lattice with self-loops. (End)

This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 0 for the central binomial sequence A000984. For a definition of Jensen polynomials see a comment in A094436. - Wolfdieter Lang, Jun 25 2019

Mirror image of triangle in A135091.

Exponential Riordan array [exp(x)*Bessel_I(0,2*x), x] = A007318 * A109187. - Peter Bala, Feb 12 2017