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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, 12, 20, 24, 15, 48, 27, 36, 16, 72, 32, 15, 40, 12, 1, 8, 14, 24, 30, 16, 18, 60, 72, 48, 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
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OFFSET
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1,2
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COMMENTS
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The Data section here is longer than usual. Do not shorten it! - N. J. A. Sloane, Jan 10 2019
The length of row n is A000041(n), the number of partitions of n.
Partitions follow the Abramowitz-Stegun (A-St) order (see the link).
The row sums give A001906(n) = Fibonacci(2*n).
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.
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.
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.
This array is the result of an attempt to understand the comment of Kevin Long, May 11 2018, on A001906.
This array is similar to A085643 but some pairs of numbers like (27, 36), (72,48), (54,144), ... are there swapped.
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LINKS
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Milton Abramowitz and Irene A. Stegun, editors, Multinomials and Partitions, Handbook of Mathematical Functions, December 1972, pp. 831-2.
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FORMULA
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EXAMPLE
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For the rows n = 1..10, and comments on compositions and set partitions with blocks of consecutive numbers, see the link.
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.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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