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A080416
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Stirling-like number triangle defined by paired decomposition of C(n+3,3) = A000292.
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1
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1, 1, 1, 1, 4, 1, 1, 12, 10, 1, 1, 32, 67, 20, 1, 1, 80, 376, 252, 35, 1, 1, 192, 1909, 2560, 742, 56, 1, 1, 448, 9094, 22928, 12346, 1848, 84, 1, 1, 1024, 41479, 189120, 177599, 46912, 4074, 120, 1, 1, 2304, 183412, 1472704, 2318149
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OFFSET
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0,5
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COMMENTS
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Note that the Stirling numbers of the second kind are generated in a similar fashion by decomposing the triangular numbers C(n+2,2) as {1}, {1,2}, {1,2,3}, .... The defining sequence A000292 appears as the subdiagonal when the triangle is arranged in lower-triangular form. The second column is A001787.
Gives the number of ways to construct pairs of k-block partitions from 1 to n such that the sum of the minima of the i-th block of the first partition and the (k-i+1)th block of the second partition is n+1. - Ken Joffaniel M Gonzales, Jun 13 2010
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LINKS
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FORMULA
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Columns are generated as follows: Display C(n+3, 3) as row sums of the triangle A080251, or {1}, {2, 2}, {3, 3, 4}, {4, 4, 6, 6}, {5, 5, 8, 8, 9}, ... The columns are then generated by 1/(1-x), 1/(1-2x)^2, 1/(1-3x)^2(1-4x)), 1/((1-4x)^2(1-6x)^2)), etc.
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EXAMPLE
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Rows are
{1},
{1, 1},
{1, 4, 1},
{1, 12, 10, 1},
{1, 32, 67, 20, 1},
...
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MATHEMATICA
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s[b_, n_, k_] := s[b, n, k] = Which[n==k==0, 1, n==0, 0, k==0, 0, True, s[b+1, n-1, k-1] + k*b*s[b, n-1, k]]
Table[s[0, n+2, k+2], {n, 0, 10}, {k, 0, n}] // Flatten
(* a specialization of equation (9) in the Corcino et al. paper *)
T[ n_, k_] := If[n < 0, 0, SeriesCoefficient[x^k / Product[1 + x*(Floor[j/2] + 1)*(Floor[j/2] - k - 1), {j, 0, k}], {x, 0, n}]]; (* Michael Somos, Oct 12 2022 *)
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PROG
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(PARI) {T(n, k) = if(n<0, 0, polcoeff(x^k / prod(j=0, k, 1 + x*(j\2 + 1)*(j\2 - k - 1) + x*O(x^n)), n))}; /* Michael Somos, Oct 12 2022 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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