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 A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038. 37
 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 3, 6, 1, 4, 6, 5, 1, 1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1, 1, 2, 2, 2, 1, 3, 6, 6, 3, 3, 4, 12, 6, 12, 1, 5, 20, 10, 6, 15, 7, 1, 1, 2, 2, 2, 2, 3, 6, 6, 3, 3, 6, 1, 4, 12, 12, 12, 12, 4, 5, 20, 10, 30, 5, 6, 30, 20, 7, 21, 8, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS This array gives in row n>=1 the multinomial numbers (call them M_0 numbers) m!/product((a_j)!,j=1..n) with the exponents of the partitions of n with number of parts m:=sum(a_j,j=1..n), given in the Abramowitz-Stegun order. See p. 831 of the given reference. See also the arrays for the M_1, M_2 and M_3 multinomial numbers A036038, A036039 and A036040 (or A080575). For a signed version see A111786. These M_0 multinomial numbers give the number of compositions of n >= 1 with parts corresponding to the partitions of n (in A-St order). See an n = 5 example below. The triangle with the summed entries of like number of parts m is A007318(n-1, m-1) (Pascal). - Wolfdieter Lang, Jan 29 2021 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20) M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972. Wolfdieter Lang, First 10 rows and more. FORMULA T(n,k) = A036040(n,k) * Factorial(A036043(n,k)) / A036038(n,k) = A049019(n,k) / A036038(n,k). If the n-th partition is P, a(n) is the multinomial coefficient of the signature of P. - Franklin T. Adams-Watters, May 30 2006 T(n,k) = A309004(A036035(n,k)). - Andrew Howroyd, Oct 19 2020 EXAMPLE Table starts:   [1, 1] [1, 2, 1] [1, 2, 1, 3, 1] [1, 2, 2, 3, 3, 4, 1] [1, 2, 2, 1, 3, 6, 1, 4, 6,  5, 1] [1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1] . T(5,6) = 4 because there are four multisets using the first four digits {0,1,2,3}: 32100, 32110, 32210 and 33210 T(5,6) = 4 because there are 4 compositions of 5 that can be formed from the partition 2+1+1+1. - Geoffrey Critzer, May 19 2013 These 4 compositions 2+1+1+1, 1+2+1+1, 1+1+2+1 and 1+1+1+2 of 5 correspond to the 4 set partitions of  :={1,2,3,4,5}, with 4 blocks of consecutive numbers, namely {1,2},{3},{4},{5} and {1},{2,3},{4},{5} and {1},{2},{3,4},{5} and {1},{2},{3},{4,5}. - Wolfdieter Lang, May 30 2018 MAPLE nmax:=9: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while st==S[k], sig))!)} Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])} { for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020 CROSSREFS Cf. A000670, A007318, A036035, A036038, A019538, A115621, A309004, A000079 (row sums), A000041 (row lengths). Sequence in context: A165357 A210961 A250007 * A111786 A072811 A296559 Adjacent sequences:  A048993 A048994 A048995 * A048997 A048998 A048999 KEYWORD nonn,tabf AUTHOR EXTENSIONS More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001 a(0)=1 prepended by Andrew Howroyd, Oct 19 2020 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)