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 A214314 Number triangle with entry T(n,m) giving the position of the first partition of n with m parts in the Abramowitz-Stegun (A-St) partition ordering. 6
 1, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 6, 7, 1, 2, 5, 8, 10, 11, 1, 2, 5, 9, 12, 14, 15, 1, 2, 6, 11, 16, 19, 21, 22, 1, 2, 6, 13, 19, 24, 27, 29, 30, 1, 2, 7, 15, 24, 31, 36, 39, 41, 42, 1, 2, 7, 17, 28, 38, 45, 50, 53, 55, 56, 1, 2, 8, 20, 35, 48, 59, 66, 71, 74, 76, 77 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For the Abramowitz-Stegun ordering of partitions see A036036 for the reference and a C. F. Hindenburg link. The present triangle is the partial sum triangle of the triangle t(n,k) = 0 if 0 <= n -1 <  k , t(n,0) = 1, n >= 1 and t(n,k) = A008284(n,k) if n-1 >=  k >= 1. This triangle with offset [1,0] for [n,k] is 1; 1,1; 1,1,1; 1,1,2,1; 1,1,2,2,1; 1,1,3,3,2,1;... (erase the diagonal of A008284 and add instead a column k=0 with only 1's). See the example section. LINKS FORMULA T(n,m) = sum(p(n,k),k=0..m-1) if n >= m >= 1, otherwise 0, with p(n,0) :=1 and p(n,k) = A008284(n,k) for k=1,2,...,n-1. EXAMPLE T(n,m) starts with: n\m   1  2  3   4   5   6   7   8   9  10  11  12... 1     1 2     1  2 3     1  2  3 4     1  2  4   5 5     1  2  4   6   7 6     1  2  5   8  10  11 7     1  2  5   9  12  14  15 8     1  2  6  11  16  19  21  22 9     1  2  6  13  19  24  27  29  30 10    1  2  7  15  24  31  36  39  41  42 11    1  2  7  17  28  38  45  50  53  55  56 12    1  2  8  20  35  48  59  66  71  74  76  77 ... T(6,4) = 8 because the 11=T(6,6) partitions for n=6 are, in A-St order: [6]; [1,5],[2,4],[3,3]; [1^2,4],[1,2,3],[2^3]; [1^3,3],[1^2,2^2]; [1^4,2]; [1^6] and the first partition with 4 parts, appears at position 8. This triangle is obtained as partial sum triangle from the triangle t(n,k) (see the comment section) which starts with: n\m   0  1  2   3   4   5   6  7  8  9 10 11 ... 1     1 2     1  1 3     1  1  1 4     1  1  2   1 5     1  1  2   2   1 6     1  1  3   3   2   1 7     1  1  3   4   3   2   1 8     1  1  4   5   5   3   2  1 9     1  1  4   7   6   5   3  2  1 10    1  1  5   8   9   7   5  3  2  1 11    1  1  5  10  11  10   7  5  3  2  1 12    1  1  6  12  15  13  11  7  5  3  2  1 ... CROSSREFS Cf. A008284. Sequence in context: A104795 A116925 A210950 * A209435 A263744 A268956 Adjacent sequences:  A214311 A214312 A214313 * A214315 A214316 A214317 KEYWORD nonn,tabl AUTHOR Wolfdieter Lang, Jul 24 2012 STATUS approved

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Last modified December 15 21:01 EST 2018. Contains 318154 sequences. (Running on oeis4.)