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 A243978 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k. 13
 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 6, 0, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 13, 1, 0, 0, 0, 0, 1, 0, 16, 4, 0, 1, 0, 0, 0, 1, 0, 25, 2, 2, 0, 0, 0, 0, 0, 1, 0, 33, 6, 1, 0, 1, 0, 0, 0, 0, 1, 0, 49, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 61, 9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 90, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 156, 9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS T(0,0) = 1 by convention. Columns k=0-10 give: A000007, A183558, A244515, A244516, A244517, A244518, A245037, A245038, A245039, A245040, A245041. Row sums are A000041. LINKS Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened) EXAMPLE Triangle starts: 00:  1; 01:  0,   1; 02:  0,   1,  1; 03:  0,   2,  0, 1; 04:  0,   3,  1, 0, 1; 05:  0,   6,  0, 0, 0, 1; 06:  0,   7,  2, 1, 0, 0, 1; 07:  0,  13,  1, 0, 0, 0, 0, 1; 08:  0,  16,  4, 0, 1, 0, 0, 0, 1; 09:  0,  25,  2, 2, 0, 0, 0, 0, 0, 1; 10:  0,  33,  6, 1, 0, 1, 0, 0, 0, 0, 1; 11:  0,  49,  4, 2, 0, 0, 0, 0, 0, 0, 0, 1; 12:  0,  61,  9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1; 13:  0,  90,  6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1; 14:  0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1; 15:  0, 156,  9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; 16:  0, 198, 23, 3, 4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1; 17:  0, 269, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; 18:  0, 334, 34, 9, 3, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1; 19:  0, 448, 27, 8, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; 20:  0, 556, 51, 7, 6, 3, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1; ... The A000041(9) = 30 partitions of 9 with the least multiplicities of any part are: 01:  [ 1 1 1 1 1 1 1 1 1 ]   9 02:  [ 1 1 1 1 1 1 1 2 ]   1 03:  [ 1 1 1 1 1 1 3 ]   1 04:  [ 1 1 1 1 1 2 2 ]   2 05:  [ 1 1 1 1 1 4 ]   1 06:  [ 1 1 1 1 2 3 ]   1 07:  [ 1 1 1 1 5 ]   1 08:  [ 1 1 1 2 2 2 ]   3 09:  [ 1 1 1 2 4 ]   1 10:  [ 1 1 1 3 3 ]   2 11:  [ 1 1 1 6 ]   1 12:  [ 1 1 2 2 3 ]   1 13:  [ 1 1 2 5 ]   1 14:  [ 1 1 3 4 ]   1 15:  [ 1 1 7 ]   1 16:  [ 1 2 2 2 2 ]   1 17:  [ 1 2 2 4 ]   1 18:  [ 1 2 3 3 ]   1 19:  [ 1 2 6 ]   1 20:  [ 1 3 5 ]   1 21:  [ 1 4 4 ]   1 22:  [ 1 8 ]   1 23:  [ 2 2 2 3 ]   1 24:  [ 2 2 5 ]   1 25:  [ 2 3 4 ]   1 26:  [ 2 7 ]   1 27:  [ 3 3 3 ]   3 28:  [ 3 6 ]   1 29:  [ 4 5 ]   1 30:  [ 9 ]   1 Therefore row n=9 is [0, 25, 2, 2, 0, 0, 0, 0, 0, 1]. MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,       b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))     end: T:= (n, k)-> b(n\$2, k) -`if`(n=0 and k=0, 0, b(n\$2, k+1)): seq(seq(T(n, k), k=0..n), n=0..14); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-i*j, i-1, k], {j, Max[1, k], n/i}]]]; T[n_, k_] := b[n, n, k] - If[n == 0 && k == 0, 0, b[n, n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *) CROSSREFS Cf. A183568, A242451 (the same for compositions). Cf. A091602 (partitions by max multiplicity of any part). Sequence in context: A243055 A318371 A245151 * A106844 A125989 A125924 Adjacent sequences:  A243975 A243976 A243977 * A243979 A243980 A243981 KEYWORD nonn,tabl AUTHOR Joerg Arndt and Alois P. Heinz, Jun 28 2014 STATUS approved

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Last modified January 20 08:54 EST 2022. Contains 350471 sequences. (Running on oeis4.)