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A205573 Array M read by antidiagonals in which successive rows evidently converge to A001405 (central binomial coefficients). 4
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 5, 1, 1, 1, 2, 3, 6, 8, 1, 1, 1, 2, 3, 6, 10, 13, 1, 1, 1, 2, 3, 6, 10, 19, 21, 1, 1, 1, 2, 3, 6, 10, 20, 33, 34, 1, 1, 1, 2, 3, 6, 10, 20, 35, 61, 55, 1, 1, 1, 2, 3, 6, 10, 20, 35, 69, 108, 89, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

CONJECTURE 1. Let M(n,k) (n,k >= 0) denote the entry in row n and column k of the array. For all n, M(n,j) = A001405(j), j=0,...,2*n+1; hence row n of M -> A001405 as n -> infinity.

Taking finite differences of even numbered columns from the top -> down yields triangle A205946 with row sums A000984, central binomial coefficients; while odd numbered columns yield triangle A205945 with row sums A001700. A205946 and A205945 represent the bisection of A191314. - Gary W. Adamson, Feb 01 2012

LINKS

Table of n, a(n) for n=0..77.

L. E. Jeffery, Unit-primitive matrices

FORMULA

Let N=2*n+3. For each n>0, define the (n+1) X (n+1) tridiagonal unit-primitive matrix (see [Jeffery]) B_n = A_{N,1} = [0,1,0,...,0; 1,0,1,0,...,0; 0,1,0,1,0,...,0; ...; 0,...,0,1,0,1; 0,...,0,1,1], and put B_0 = [1]. Then, for all n, M(n,k)=[(B_n)^k]_{n+1,n+1}, k=0,1,..., where X_{n+1,n+1} denotes the lower right corner entry of X.

CONJECTURE 2 (Rows of M). Let S(n,i) denote term i in row n of A115139, i=0,...,floor(n/2), and let T(n,j) denote term j in row n of A108299, j=0,...,n.  The generating function for row n of M is of the form F_n(x) =sum[i=0,...,floor(n/2) S(n,i)*x^(2*i)]/sum[j=0,...,n T(n,j)*x^j].

CONJECTURE 3 (Columns of M). Let D(m,k) denote term m in column k of A191314, m=0,...,floor(k/2). The generating function for column k of M is of the form G_k(x)=sum[m=0,...,floor(k/2) D(m,k)*x^m]/(1-x).

EXAMPLE

Array begins

1, 1, 1, 1, 1,  1,  1,  1,  1,   1,   1,...

1, 1, 2, 3, 5,  8, 13, 21, 34,  55,  89,...

1, 1, 2, 3, 6, 10, 19, 33, 61, 108, 197,...

1, 1, 2, 3, 6, 10, 20, 35, 69, 124, 241,...

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 251,...

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252,...

...

According to Conjecture 2, row n=3 has g.f. F_3(x)=(1-2*x^2)/(1-x-3*x^2+2*x^3+x^4).

CROSSREFS

Cf. A001405, A108299, A115139, A191314.

Cf. also A205945, A205946, A001700, A000984.

Sequence in context: A245563 A122945 A209972 * A119338 A054124 A144406

Adjacent sequences:  A205570 A205571 A205572 * A205574 A205575 A205576

KEYWORD

nonn,tabl

AUTHOR

L. Edson Jeffery, Jan 29 2012

STATUS

approved

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Last modified August 13 08:27 EDT 2020. Contains 336442 sequences. (Running on oeis4.)