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 A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle). 3
 1, 1, 1, 3, 1, 1, 15, 3, 1, 1, 1, 105, 15, 3, 3, 1, 1, 1, 945, 105, 15, 9, 15, 3, 1, 3, 1, 1, 1, 10395, 945, 105, 45, 105, 15, 9, 3, 15, 3, 1, 3, 1, 1, 1, 135135, 10395, 945, 315, 225, 945, 105, 45, 15, 9, 105, 15, 9, 3, 1, 15, 3, 1, 3, 1, 1, 1, 2027025, 135135, 10395, 2835 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(-1;n,k) with the k-th partition of n in A-St order. The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...]. If M32hat(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-1):= A144270(n,m). LINKS W. Lang, First 10 rows of the array and more. W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3. FORMULA a(n,k)= product(|S2(-1,j,1)|^e(n,k,j),j=1..n) with |S2(-1,n,1)|= A001147(n-1) = (2*n-3)(!^2) (2-factorials) for n>=2 and 1 if n=1 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. Formally a(n,k)= 'M32(-1)/M3' = 'A143171/A036040' (elementwise division of arrays). EXAMPLE a(4,3)= 1 = |S2(-1,2,1)|^2. The relevant partition of 4 is (2^2). [1]; [1,1]; [3,1,1]; [15,3,1,1,1]; [105,15,3,3,1,1,1]; ... [From Wolfdieter Lang, Oct 23 2008] CROSSREFS A144271 (M32hat(-2) array). Sequence in context: A073483 A006956 A072285 * A144270 A110112 A326800 Adjacent sequences:  A144266 A144267 A144268 * A144270 A144271 A144272 KEYWORD nonn,easy,tabf AUTHOR Wolfdieter Lang Oct 09 2008 EXTENSIONS Corrected all entries. Wolfdieter Lang, Oct 23 2008 STATUS approved

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Last modified August 10 16:56 EDT 2020. Contains 336381 sequences. (Running on oeis4.)