cp's OEIS Frontend

The denominators are A130188.

The z-sequence for the Sheffer matrix (see the W. Lang link under A006233) A060821(n,m) (coefficients of Hermite polynomials) is z(2*k)=0 and z(2*k+1) = -r(k)/2, k >= 0, with r(n) := a(n)/A130188(n).

The recurrence for the entries of the first (m=0) column of the Sheffer triangle A006233(n,m)=:H(n,m) is H(0,0):=1, H(n,0) = n*Sum_{m=0..n-1} z(m)*H(n-1, m), n >= 1.

The e.g.f. for the z-sequence is -2*(exp((x^2)/4)-1)/x.

Numerators are A130187.

See A130187 for details on r(n):= A130187(n)/a(n) and the corresponding z-sequence for the Hermite matrix A060821.

Row sums are:

{1, 2, -6, -52, -116, 312, 2584, 1808, -42864, -144352, 601504};

As an operator algebra like an Energy Hamiltonian:

e(n)*H(x,n)=p(x,n)/x^2

The relative energy of the row sums goes up much faster than in the Chebyshev

of the first kind.

Row sums are:

{1, 2, 2, -4, -20, -8, 184, 464, -1648, -10720, 8224}

In real quantum mechanical 2 dimensional orthogonal partitions it would be:

p(x,y,n,m)=T(x,n)*H(y,m).

Here I have made x=y and n=m to get a new sort of polynomial with an odd number of vector coefficients.

The traditional Schoedinger wave mechanics solution of hydrogen is a partition of four (not two dimensions): wave_function=Bessel(r,n)*Legendre(theta,l)*Fourier(phi,m)*Spin(t,s).

Row sums are:

{1, 4, 16, 68, 332, 1952, 13648, 109552, 986896, 9865664, 108500864};

Row sums are:

{3, 6, 6, -12, -60, -24, 552, 1392, -4944, -32160, 24672}

Implicit surface of:

f[x_, y_, z_] = 128 x^6 - 480 x^4 y^2 + 720 x^2 y^4 - 56 y^6 - 480 x^4 z^2 - 480y^4 z^2 + 720 x^2 z^4 + 720 y^2 z^4 - 240 z^6 - 1

has twelve horns off the main body ( more than the Chebyshev at the same level).

Number of solutions to the rook problem on a 2n X 2n board having a certain symmetry group (see Robinson for details).

Also the value of the n-th derivative of exp(x^2) evaluated at 1. - N. Calkin, Apr 22 2010

For n >= 1, a(n) is also the sum of the degrees of the irreducible representations of the group of n X n signed permutation matrices (described in sequence A066051). The similar sum for the "ordinary" symmetric group S_n is in sequence A000085. - Sharon Sela (sharonsela(AT)hotmail.com), Jan 12 2002

It appears that this is also the number of permutations of 1, 2, ..., n+1 such that each term (after the first) is within 2 of some preceding term. Verified for n+1 <= 6. E.g., a(4) = 20 because of the 24 permutations of 1, 2, 3, 4, the only ones not permitted are 1, 4, 2, 3; 1, 4, 3, 2; 4, 1, 2, 3; and 4, 1, 3, 2. - Gerry Myerson, Aug 06 2003

Hankel transform is A108400. - Paul Barry, Feb 11 2008

From Emeric Deutsch, Jun 19 2010: (Start)

Number of symmetric involutions of [2n]. Example: a(2)=6 because we have 1234, 2143, 1324, 3412, 4231, and 4321. See the Egge reference, pp. 419-420.

Number of symmetric involutions of [2n+1]. Example: a(2)=6 because we have 12345, 14325, 21354, 45312, 52341, and 54321. See the Egge reference, pp. 419-420.

(End)

Binomial convolution of sequence A000085: a(n) = Sum_{k=0..n} binomial(n,k)*A000085(k)*A000085(n-k). - Emanuele Munarini, Mar 02 2016

The sequence can be obtained from the infinite product of 2 X 2 matrices [(1,N); (1,1)] by extracting the upper left terms, where N = (1, 3, 5, ...), the odd integers. - Gary W. Adamson, Jul 28 2016

Apparently a(n) is the number of standard domino tableaux of size 2n, where a domino tableau is a generalized Young tableau in which all rows and columns are weakly increasing and all regions are dominos. - Gus Wiseman, Feb 25 2018