cp's OEIS Frontend

A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.

%I A091355 #25 Jan 23 2016 08:33:08
%S A091355 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 A091355 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 A091355 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 A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.
%C A091355 Row sums give A000219.
%C A091355 Columns 1-5 are respectively A000041, A091356, A091357, A091358, and A091359.
%C A091355 Columns converge to A091360.
%H A091355 Alois P. Heinz, <a href="/A091355/b091355.txt">Rows n = 1..141, flattened</a>
%F A091355 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 A091355 Triangle starts:
%e A091355 01:  1,
%e A091355 02:  2, 1,
%e A091355 03:  3, 2, 1,
%e A091355 04:  5, 5, 2, 1,
%e A091355 05:  7, 9, 5, 2, 1,
%e A091355 06:  11, 18, 11, 5, 2, 1,
%e A091355 07:  15, 30, 22, 11, 5, 2, 1,
%e A091355 08:  22, 53, 42, 24, 11, 5, 2, 1,
%e A091355 09:  30, 85, 78, 46, 24, 11, 5, 2, 1,
%e A091355 10:  42, 139, 138, 90, 48, 24, 11, 5, 2, 1,
%e A091355 11:  56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1,
%e A091355 12:  77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1,
%e A091355 13:  101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1,
%e A091355 14:  135, 760, 1088, 899, 580, 336, 182, 96, 48, 24, 11, 5, 2, 1,
%e A091355 15:  176, 1115, 1741, 1512, 1020, 606, 340, 182, 96, 48, 24, 11, 5, 2, 1,
%e A091355 ...
%p A091355 with(numtheory):
%p A091355 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
%p A091355       min(d, k)*d, d=divisors(j))*A(n-j, k), j=1..n)/n)
%p A091355     end:
%p A091355 T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
%p A091355 seq(seq(T(n, k), k=1..n), n=1..15);  # _Alois P. Heinz_, Mar 15 2014
%t A091355 (* 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 A091355 (* second program: *)
%t A091355 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 A091355 nonn,tabl
%O A091355 1,2
%A A091355 _Christian G. Bower_, Jan 02 2004
%E A091355 Definition corrected, _Joerg Arndt_, Jul 21 2014