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A117433 Number of planar partitions of n with all part sizes distinct. 2


%S 1,1,1,3,3,5,9,11,15,21,35,41,59,75,103,149,187,243,321,413,527,735,

%T 895,1165,1467,1885,2335,2997,3853,4765,5977,7473,9269,11531,14255,

%U 17537,22201,26897,33233,40613,50027,60637,74459,89963,109751,134407,162117

%N Number of planar partitions of n with all part sizes distinct.

%C Matches A072706 for n < 10, since a unimodal composition into distinct parts can be placed uniquely as a hook. Starting with n = 10, additional partitions are possible (starting with [4,3|2,1] and [4,2|3,1]).

%H Franklin T. Adams-Watters and Alois P. Heinz, <a href="/A117433/b117433.txt">Table of n, a(n) for n = 0..1000</a> (first 100 terms from Franklin T. Adams-Watters)

%F a(n) = sum_{k=1}^{floor((sqrt(8*n+1)-1)/2)} A000085(k)*A008289(n,k).

%p b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)

%p -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))

%p end:

%p g:= proc(n) g(n):= `if`(n<2, 1, (n-1)*g(n-2) +g(n-1)) end:

%p a:= proc(n) b(n, n); add(%[i]*g(i-1), i=1..nops(%)) end:

%p seq (a(n), n=0..60); # _Alois P. Heinz_, Nov 18 2012

%Y Cf. A000219, A072706, A117434, A000009.

%K nonn

%O 0,4

%A _Franklin T. Adams-Watters_, Mar 16 2006, Apr 01 2008

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Last modified December 12 01:08 EST 2017. Contains 295936 sequences.