cp's OEIS Frontend

A305309 Array read by rows: a(n, k) = A048996(n, k) * A118851(n, k), n >= 1, k = 1..A000041(n).

%I A305309 #27 Jan 11 2019 04:21:58
%S A305309 1,2,1,3,4,1,4,6,4,6,1,5,8,12,9,12,8,1,6,10,16,9,12,36,8,12,24,10,1,7,
%T A305309 12,20,24,15,48,27,36,16,72,32,15,40,12,1,8,14,24,30,16,18,60,72,48,
%U A305309 54,20,96,54,144,16,20,120,80,18,60,14,1,9,16,28,36,40,21,72,90,48,60,144,27,24,120,144,192,216,96,25,160,90,360,80,24,180,160,21,84,16,1,10,18,32,42,48,25,24,84,108,120,72,180,96,108,28,144,180,96,240,576,108,128,216,30,200,240,480,540,480,32,30,240,135,720,240,28,252,280,24,112,18,1
%N A305309 Array read by rows: a(n, k) = A048996(n, k) * A118851(n, k), n >= 1, k = 1..A000041(n).
%C A305309 The Data section here is longer than usual.  Do not shorten it! - _N. J. A. Sloane_, Jan 10 2019
%C A305309 The length of row n is A000041(n), the number of partitions of n.
%C A305309 Partitions follow the Abramowitz-Stegun (A-St) order (see the link).
%C A305309 The row sums give A001906(n) = Fibonacci(2*n).
%C A305309 The triangle T(n, m) obtained by summing in row n the entries of the columns k with identical part number m is A078812(n, m) = binomial(n+m-1, 2*m-1) (with offsets n >= 1, m = 1..n). The array of the number of parts m = m(n,k) = A036043(n, k) in A-St order.
%C A305309 This array is the elementwise product of the array A048996, the composition numbers, and A118851, the products of the parts of partitions, both arrays are in A-St order.
%C A305309 Therefore a(n, k) is the sum of the number of products of the block lengths of all the A048996(n, k) set partition of [n] := {1,2, ..., n} with m = m(n, k) blocks consisting of consecutive numbers corresponding to the k-th partition of n in A-St order. Because the block structure depends only on the exponents (signature) of the underlying partition this leads to the product of the two array entries. Equivalently, one can consider compositions. Then a(n, k) gives the sum of the products of the parts of all A048996(n, k) compositions originating from the k-th partition of n.
%C A305309 This array is the result of an attempt to understand the comment of _Kevin Long_, May 11 2018, on A001906.
%C A305309 This array is similar to A085643 but some pairs of numbers like  (27, 36), (72,48), (54,144), ... are there swapped.
%H A305309 Milton Abramowitz and Irene A. Stegun, editors, <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=831&amp;Submit=Go">Multinomials and Partitions</a>, Handbook of Mathematical Functions, December 1972, pp. 831-2.
%H A305309 Wolfdieter Lang, <a href="/A305309/a305309.pdf">Rows n = 1..10, and more.</a>
%F A305309 a(n, k) =  A048996(n, k) * A118851(n, k), n >= 1, k = 1..A000041(n).
%e A305309 For the rows n = 1..10, and comments on compositions and set partitions with blocks of consecutive numbers, see the link.
%e A305309 Example: n = 5, k = 4: the partition is (1^2, 3^1) = [1,1,3] with m = m(n,k) = 3. The A048996(5, 4) = 3 compositions are 1 + 1 + 3, 1 + 3 + 1 and 3 + 1 + 1. The corresponding three consecutive 3-block partitions of [5] := {1, 2, ..., 5} are {1}, {2}, {3,4,5} and {1}, {2,3,4}, {5} and {1,2,3}, {4}, {5}, Therefore, a(5, 4) = 1*1*3 + 1*3*1 + 3*1*1 = 3*3 = 9. For the compositions one has the same sum from the products of the parts.
%Y A305309 Cf. A000041, A001906, A036043, A048996, A078812, A085643, A118851.
%K A305309 nonn,tabf
%O A305309 1,2
%A A305309 _Wolfdieter Lang_, May 31 2018