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 A213008 Triangle of number of distinct values of multinomial coefficients corresponding to sequence A026820. 2
 1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 5, 6, 7, 1, 4, 7, 9, 10, 11, 1, 4, 8, 10, 12, 13, 14, 1, 5, 9, 14, 16, 18, 19, 20, 1, 5, 12, 17, 21, 23, 25, 26, 27, 1, 6, 13, 21, 26, 30, 32, 34, 35, 36, 1, 6, 16, 25, 33, 37, 41, 43, 45, 46, 47, 1, 7, 19, 32, 42, 50, 54, 58, 60, 62, 63, 64 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Differs from A026820 after position 24. Includes sequence A070289 when k=n. LINKS Alois P. Heinz, Rows n = 1..45, flattened Katsuhisa Yamanaka, Shin-ichiro Kawano, Yosuke Kikuchi, Shin-ichi Nakano, Constant Time Generation of Integer Partitions, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol.E90-A, no.5, pp.888-895, (May-2007). Sergei Viznyuk, C-Program, same, local copy. EXAMPLE Triangle begins: 1; 1, 2; 1, 2, 3; 1, 3, 4,  5; 1, 3, 5,  6,  7; 1, 4, 7,  9, 10, 11; 1, 4, 8, 10, 12, 13, 14; Thus, for n=7 and k=6 there are 13 distinct values of multinomial coefficients corresponding to partitions of n=7 into at most k=6 parts. The corresponding number of partitions from sequence A026820 is 14. That is because partitions 7=4+1+1+1 and 7=3+2+2 produce the same value of multinomial coefficient 7!/(4!*1!*1!*1!)=7!/(3!*2!*2!). MAPLE b:= proc(n, i, k) option remember; if n=0 then {1} elif i<1       then {} else {b(n, i-1, k)[], seq(map(x-> x*i!^j,               b(n-i*j, i-1, k-j))[], j=1..min(n/i, k))} fi     end: T:= (n, k)-> nops(b(n, n, k)): seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 14 2012 MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1}, If[i<1, {}, Join[b[n, i-1, k], Table[ Function[#*i!^j] /@ b[n-i*j, i-1, k-j], {j, 1, Min[n/i, k]}] // Flatten] // Union] ]; T[n_, k_] := Length[b[n, n, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *) CROSSREFS Cf. A026820, A070289. Sequence in context: A066010 A209556 A109974 * A215520 A026820 A091438 Adjacent sequences:  A213005 A213006 A213007 * A213009 A213010 A213011 KEYWORD nonn,tabl AUTHOR Sergei Viznyuk, Jun 01 2012 STATUS approved

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