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%I #25 Jan 23 2016 08:33:08
%S 1,2,1,3,2,1,5,5,2,1,7,9,5,2,1,11,18,11,5,2,1,15,30,22,11,5,2,1,22,53,
%T 42,24,11,5,2,1,30,85,78,46,24,11,5,2,1,42,139,138,90,48,24,11,5,2,1,
%U 56,215,239,164,94,48,24,11,5,2,1,77,336,405,298,176,96,48,24,11,5,2,1,101,504,669,520,324,180,96,48,24,11,5,2,1
%N Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.
%C Row sums give A000219.
%C Columns 1-5 are respectively A000041, A091356, A091357, A091358, and A091359.
%C Columns converge to A091360.
%H Alois P. Heinz, <a href="/A091355/b091355.txt">Rows n = 1..141, flattened</a>
%F k-th column is EulerTransform[1, 2, 3, .., k, k, k, ..]-EulerTransform[1, 2, 3, .., k-1, k-1, k-1, ..]. - _Wouter Meeussen_, Aug 29 2004
%e Triangle starts:
%e 01: 1,
%e 02: 2, 1,
%e 03: 3, 2, 1,
%e 04: 5, 5, 2, 1,
%e 05: 7, 9, 5, 2, 1,
%e 06: 11, 18, 11, 5, 2, 1,
%e 07: 15, 30, 22, 11, 5, 2, 1,
%e 08: 22, 53, 42, 24, 11, 5, 2, 1,
%e 09: 30, 85, 78, 46, 24, 11, 5, 2, 1,
%e 10: 42, 139, 138, 90, 48, 24, 11, 5, 2, 1,
%e 11: 56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1,
%e 12: 77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1,
%e 13: 101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1,
%e 14: 135, 760, 1088, 899, 580, 336, 182, 96, 48, 24, 11, 5, 2, 1,
%e 15: 176, 1115, 1741, 1512, 1020, 606, 340, 182, 96, 48, 24, 11, 5, 2, 1,
%e ...
%p with(numtheory):
%p A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
%p min(d, k)*d, d=divisors(j))*A(n-j, k), j=1..n)/n)
%p end:
%p T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
%p seq(seq(T(n, k), k=1..n), n=1..15); # _Alois P. Heinz_, Mar 15 2014
%t (* load EulerTransform from 'seqtranslib.m' under OEIS-Transforms *) Table[EulerTransform[Table[Min[c, r], {r, 20}]] - EulerTransform[Table[Min[c-1, r], {r, 20}]], {c, 20}] // Transpose
%t (* second program: *)
%t A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k]*d, {d, Divisors[j]}] *A[n-j, k], {j, 1, n}]/n]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1] ]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* _Jean-François Alcover_, Jan 23 2016, after _Alois P. Heinz_ *)
%K nonn,tabl
%O 1,2
%A _Christian G. Bower_, Jan 02 2004
%E Definition corrected, _Joerg Arndt_, Jul 21 2014
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