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A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.

%I A227551 #33 Jan 11 2019 08:44:06
%S A227551 1,0,1,0,1,0,1,1,0,1,1,0,1,2,0,1,3,0,1,3,1,0,1,3,2,0,1,5,2,0,1,5,4,0,
%T A227551 1,5,6,0,1,6,7,1,0,1,6,10,1,0,1,7,11,3,0,1,9,13,4,0,1,7,18,6,0,1,8,20,
%U A227551 9,0,1,10,21,14,0,1,9,27,16,1,0,1,10,29,22,2
%N A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.
%C A227551 The boundary size is the number of parts having fewer than two neighbors.
%H A227551 Alois P. Heinz, <a href="/A227551/b227551.txt">Rows n = 0..600, flattened</a>
%e A227551 T(12,1) = 1: [12].
%e A227551 T(12,2) = 6: [1,11], [2,10], [3,4,5], [3,9], [4,8], [5,7].
%e A227551 T(12,3) = 7: [1,2,3,6], [1,2,9], [1,3,8], [1,4,7], [1,5,6], [2,3,7], [2,4,6].
%e A227551 T(12,4) = 1: [1,2,4,5].
%e A227551 Triangle T(n,k) begins:
%e A227551   1;
%e A227551   0, 1;
%e A227551   0, 1;
%e A227551   0, 1, 1;
%e A227551   0, 1, 1;
%e A227551   0, 1, 2;
%e A227551   0, 1, 3;
%e A227551   0, 1, 3, 1;
%e A227551   0, 1, 3, 2;
%e A227551   0, 1, 5, 2;
%e A227551   0, 1, 5, 4;
%e A227551   0, 1, 5, 6;
%e A227551   0, 1, 6, 7, 1;
%p A227551 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
%p A227551       expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
%p A227551       `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
%p A227551     end:
%p A227551 T:= n-> (p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
%p A227551 seq(T(n), n=0..30);
%t A227551 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]]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 30}] // Flatten (* _Jean-François Alcover_, Dec 12 2016, after _Alois P. Heinz_ *)
%Y A227551 Columns k=0-10 give: A000007, A057427, A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567.
%Y A227551 Row sums give: A000009.
%Y A227551 Last elements of rows give: A227552.
%Y A227551 Cf. A227345 (a version with trailing zeros), A053993, A201077, A227568, A224878 (one part of size 0 allowed).
%K A227551 nonn,look,tabf
%O A227551 0,14
%A A227551 _Alois P. Heinz_, Jul 16 2013