cp's OEIS Frontend

Previous name was: Invert transform of right-shifted partition function (A000041).

Sums of the antidiagonals of the array formed by sequences A000007, A000041, A000712, A000716, ... or its transpose A000012, A000027, A000096, A006503, A006504, ....

Row sums of triangle A143866 = (1, 2, 5, 12, 29, 69, 165, ...) and right border of A143866 = (1, 1, 2, 5, 12, ...). - Gary W. Adamson, Sep 04 2008

Starting with offset 1 = A137682 / A000041; i.e. (1, 3, 7, 17, 40, 96, ...) / (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 01 2009

From L. Edson Jeffery, Mar 16 2011: (Start)

Another approach is the following. Let T be the infinite lower triangular matrix with columns C_k (k=0,1,2,...) such that C_0=A000041 and, for k > 0, such that C_k is the sequence giving the number of partitions of n into parts of k+1 kinds (successive self-convolutions of A000041 yielding A000712, A000716, ...) and shifted down by k rows. Then T begins (ignoring trailing zero entries in the rows)

(1, 0, ... )

(1, 1, 0, ... )

(2, 2, 1, 0, ... )

(3, 5, 3, 1, 0, ... )

(5, 10, 9, 4, 1, 0, ...)

etc., and a(n) is the sum of entries in row n of T. (End)

This infinite lower triangular Riordan matrix T is the so-called L-eigen-matrix of the infinite lower triangular Riordan matrix A027293 (but with offset 0 for rows and columns). Such eigentriangles have been considered by Paul Barry in the paper given as a link in A186020.

This means that E is the L-eigen-matrix of an infinite lower triangular matrix M if M*E = L*(E - I), with the unit matrix I and the matrix L with elements L(i, j) = delta_{i, j-1} (Kronecker's delta-symbol; first upper diagonal with 1's).

Therefore, this notion is analogous to calling sequence S an L-eigen-sequence of matrix M if M*vec(S) = L.vec(S) (or vec(S) is an eigensequence of M - L with eigenvalue 0), used by Bernstein and Sloane, see the links in A155002.

L*(E - I) is the E matrix after elimination of the main diagonal and then the first row, and starting with offset 0. Because for infinite lower triangular matrices L^{tr}.L = I (tr stands for transposed), this leads to M = L*(I - E^{-1}) or E = (I - L^{tr}*M)^{-1}.

Note that Gary W. Adamson uses a different notion: E is the eigentriangle of a triangle T if the columns of E are the columns j of T multiplied by the sequence elements B_j of B with o.g.f. x/(1 - x*G(x)), with the o.g.f. G(x) of column no. 1 of T. Or E(i, j) = T(i, j)*B(j). In short: sequence B is the L-eigen-sequence of the infinite lower triangular matrix T (but with offset 1): T*vec(B) = L.vec(B). See, e.g., A143866.

Thanks to Gary W. Adamson for motivating my occupation with such eigentriangles and eigensequences.

The first column of the present triangle T is A067687, which is then shifted downwards (Riordan of Toeplitz type).