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A227552 Number of partitions of n into distinct parts with maximal boundary size.

%I A227552 #14 Feb 15 2017 09:27:05
%S A227552 1,1,1,1,1,2,3,1,2,2,4,6,1,1,3,4,6,9,14,1,2,3,5,8,11,17,24,1,1,3,5,8,
%T A227552 11,18,24,35,49,1,2,3,6,9,14,21,30,42,60,81,1,1,3,5,9,13,21,29,43,60,
%U A227552 84,113,156,1,2,3,6,10,15,24,35,50,71,99,134,184,246
%N A227552 Number of partitions of n into distinct parts with maximal boundary size.
%C A227552 The boundary size is the number of parts having less than two neighbors.
%H A227552 Alois P. Heinz, <a href="/A227552/b227552.txt">Table of n, a(n) for n = 0..1000</a>
%F A227552 a(n) = A227551(n,A227568(n)).
%p A227552 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
%p A227552       expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
%p A227552       `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
%p A227552     end:
%p A227552 a:= n-> (p->coeff(p, x, degree(p)))(b(n$2, 0)):
%p A227552 seq(a(n), n=0..100);
%t A227552 b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t>1, x, 1], Expand[If[i<1, 0, If[t>1, x, 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t==2, x, 1] * b[n-i, i-1, Quotient[t, 2]+2]]]]]; a[n_] := Function [p, Coefficient[p, x, Exponent[p, x]]][b[n, n, 0]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 15 2017, translated from Maple *)
%Y A227552 Last elements of rows of A227551.
%Y A227552 Last nonzero elements of rows of A227345.
%Y A227552 Cf. A053993, A201077.
%K A227552 nonn
%O A227552 0,6
%A A227552 _Alois P. Heinz_, Jul 16 2013