This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A227345 #32 May 15 2025 21:52:47
%S A227345 1,1,0,1,1,0,1,1,0,0,1,2,0,0,0,1,3,0,0,0,0,1,3,1,0,0,0,0,1,3,2,0,0,0,
%T A227345 0,0,1,5,2,0,0,0,0,0,0,1,5,4,0,0,0,0,0,0,0,1,5,6,0,0,0,0,0,0,0,0,1,6,
%U A227345 7,1,0,0,0,0,0,0,0,0,1,6,10,1,0,0,0,0,0,0,0,0,0,1,7,11,3,0,0,0,0,0,0,0,0,0,0,1,9,13,4,0
%N A227345 Triangle read by rows, partitions into distinct parts by size of boundary.
%C A227345 The boundary size of a partition is the number of parts p that do not have two neighbors (that is, not both p-1 and p+1 are parts).
%C A227345 Row sums are A000009.
%C A227345 Conjecture: there exists a partition (into distinct parts) of n with boundary size k if and only if 0 < k^2 * 3/4 <= n. - _Patrick Devlin_, Jul 13 2013
%H A227345 Joerg Arndt, <a href="/A227345/b227345.txt">Table of n, a(n) for n = 1..5050</a>
%F A227345 From _Patrick Devlin_, Jul 13 2013: (Start)
%F A227345 Let a(n,k) denote the number of partitions into distinct parts of n with boundary size k. Then for all n>0 and k>=0, we have a(n,k+1) >= floor(binomial(n-k, k) * 2^(-binomial(k, 2))) = floor(binomial(n-k, k) * 2^(-A000217(k))). (Proof is by noting a(n,k) >= Sum_{j=1..(n/2-1)} a(j,k-1).)
%F A227345 On the other hand, for all n>0 and k>=0, we also have that a(n,k+1) <= binomial(n-k,k)*A000045(k+1). This is obtained by considering the largest k parts of the boundary, which must be some subset of {1, 2, ..., n-k}. Then the possible 'gaps' of the boundary can each either be filled with the corresponding consecutive integers or left empty. (End)
%e A227345 Triangle starts (dots for zeros, trailing zeros omitted for n>=14):
%e A227345 01: 1
%e A227345 02: 1 .
%e A227345 03: 1 1 .
%e A227345 04: 1 1 . .
%e A227345 05: 1 2 . . .
%e A227345 06: 1 3 . . . .
%e A227345 07: 1 3 1 . . . .
%e A227345 08: 1 3 2 . . . . .
%e A227345 09: 1 5 2 . . . . . .
%e A227345 10: 1 5 4 . . . . . . .
%e A227345 11: 1 5 6 . . . . . . . .
%e A227345 12: 1 6 7 1 . . . . . . . .
%e A227345 13: 1 6 10 1 . . . . . . . . .
%e A227345 14: 1 7 11 3 . . . . . . . . .
%e A227345 15: 1 9 13 4 . . . . . . . . .
%e A227345 16: 1 7 18 6 . . . . . . . . .
%e A227345 17: 1 8 20 9 . . . . . . . . .
%e A227345 18: 1 10 21 14 . . . . . . . .
%e A227345 19: 1 9 27 16 1 . . . . . . .
%e A227345 20: 1 10 29 22 2 . . . . . . .
%e A227345 21: 1 12 32 28 3 . . . . . . .
%e A227345 22: 1 11 37 35 5 . . . . . . .
%e A227345 23: 1 11 42 42 8 . . . . . . .
%e A227345 24: 1 12 45 53 11 . . . . . .
%e A227345 25: 1 13 49 62 17 . . . . . .
%e A227345 26: 1 13 54 73 24 . . . . . .
%e A227345 27: 1 15 58 86 31 1 . . . . .
%e A227345 28: 1 14 65 98 43 1 . . . . .
%e A227345 29: 1 14 70 114 54 3 . . . . .
%e A227345 30: 1 17 72 134 67 5 . . . . .
%e A227345 31: 1 15 82 148 86 8 . . . . .
%e A227345 32: 1 15 87 168 108 11 . . . .
%e A227345 33: 1 18 90 192 129 18 . . . .
%e A227345 34: 1 17 98 212 160 24 . . . .
%e A227345 35: 1 19 103 235 192 35 . . .
%e A227345 36: 1 19 111 264 224 49 . . .
%e A227345 37: 1 18 119 289 268 64 1 . .
%e A227345 38: 1 19 124 320 315 83 2 . .
%e A227345 39: 1 21 130 355 360 112 3 . .
%e A227345 40: 1 20 139 385 424 138 6 . .
%e A227345 In particular, for the tenth row of this table, note that the partitions of ten into distinct parts are 10 = 10 = 9 + 1 = 8 + 2 = 7 + 3 = 6 + 4 = 4 + 3 + 2 + 1 = 7 + 2 + 1 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2. These partitions are sorted by increasing number of parts in the boundary. In particular, note that 4 + 3 + 2 + 1 has only two parts in its boundary (namely 4 and 1). - _Patrick Devlin_, Jul 13 2013
%p A227345 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
%p A227345 expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
%p A227345 `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
%p A227345 end:
%p A227345 T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
%p A227345 seq(T(n), n=1..20); # _Alois P. Heinz_, Jul 16 2013
%t A227345 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, 1, n}]][b[n, n, 0]]; Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Feb 18 2015, after _Alois P. Heinz_ *)
%Y A227345 Cf. A227344 (partitions by perimeter).
%Y A227345 Columns k=1-10 give: A057427 (for n>=1), A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567. Cf. A227551 (a version without trailing zeros), A227552. - _Alois P. Heinz_, Jul 16 2013
%K A227345 nonn,tabl
%O A227345 1,12
%A A227345 _Joerg Arndt_, Jul 08 2013