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 A300185 Irregular triangle read by rows: T(n, {j,k}) is the number of partitions of n that have exactly j parts equal to k; 1 <= j <= n, 1 <= k <= n. 1
 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 1, 1, 0, 1, 3, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 2, 2, 1, 1, 0, 1, 4, 2, 1, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 3, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,31 COMMENTS Row sums = A027293. If superfluous zeros are removed from the right side of each row, the row lengths = 1,2,1,3,1,1,4,2,... = A010766. Sum of each N X N block of rows = 1,2,4,7,12,19,... = A000070. The sum of the partitions of n that are over-counted in each block of N x N rows = A000070(n) - A000041(n) = A058884(n), n >= 1. Concatenation of first row from each N X N block = A116598. As noted by Joerg Arndt in A116598, the first row from each N X N block in reverse converges to A002865. Two sequences emerge from alternating second rows in reverse: for 2n, converges to even-indexed terms in A027336, and for 2n+1, converges to odd-indexed terms in A027336. Counting the rows in each N X N block where columns j=2 > 0 and j=3 through j=n are all zeros produces A008615(n), n > 0. LINKS J. Stauduhar, Table of n, a(n) for n = 1..10000 Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014. J. Stauduhar, Original Python program. EXAMPLE \ j 1 2 3 4 5 k n 1: 1 1 2: 1 0 1 2 1 0 3: 1 1 0 1 2 1 0 0 3 1 0 0 4: 1 1 1 0 1 2 1 1 0 0 3 1 0 0 0 4 1 0 0 0 5: 1 2 1 1 0 1 2 2 1 0 0 0 3 2 0 0 0 0 4 1 0 0 0 0 5 1 0 0 0 0 . . . MATHEMATICA Array[With[{s = IntegerPartitions[#]}, Table[Count[Map[Count[#, k] &, s], j], {k, #}, {j, #}]] &, 7] // Flatten (* Michael De Vlieger, Feb 28 2018 *) PROG (Python) # See Stauduhar link. CROSSREFS Cf. A000041, A000070, A008615, A010766, A027293, A027336, A058884, A116598. Sequence in context: A321519 A346392 A169590 * A262804 A048825 A339884 Adjacent sequences: A300182 A300183 A300184 * A300186 A300187 A300188 KEYWORD nonn,tabf AUTHOR J. Stauduhar, Feb 27 2018 STATUS approved

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Last modified September 18 01:15 EDT 2024. Contains 375995 sequences. (Running on oeis4.)