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 A134273 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5). 4
 1, 5, 1, 45, 15, 1, 585, 180, 75, 30, 1, 9945, 2925, 2250, 450, 375, 50, 1, 208845, 59670, 43875, 20250, 8775, 13500, 1875, 900, 1125, 75, 1, 5221125, 1461915, 1044225, 921375, 208845, 307125, 141750, 118125, 20475, 47250, 13125, 1575, 2625, 105, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506. Partition number array M_3(5), the k=5 member in the family of a generalization of the multinomial number arrays M_3 = M_3(1) = A036040. The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...]. The S2(5,n,m):=A049029(n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3 = A036040. a(n,k) enumerates unordered forests of increasing quintic (5-ary) trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition. LINKS M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. W. Lang, First 10 rows and more. FORMULA a(n,k) = n!*Product_{j=1..n} (S2(5,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-factorials) and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. EXAMPLE ; ; [45,15,1]; [585,180,75,30,1]; [9945,2925,2250,450,375,50,1]; ... CROSSREFS Cf. There are a(4, 3)=75=3*5^2 unordered 2-forest with 4 vertices, composed of two 5-ary increasing trees, each with two vertices: there are 3 increasing labelings (1, 2)(3, 4); (1, 3)(2, 4); (1, 4)(2, 3) and each tree comes in five versions from the 5-ary structure. Cf. A049120 (row sums also of triangle A049029). Cf. A134149 (M_3(4) array). Sequence in context: A264774 A114154 A297899 * A048897 A049029 A051150 Adjacent sequences:  A134270 A134271 A134272 * A134274 A134275 A134276 KEYWORD nonn,easy,tabf AUTHOR Wolfdieter Lang Nov 13 2007 STATUS approved

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Last modified February 19 19:06 EST 2020. Contains 332047 sequences. (Running on oeis4.)