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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. 14
 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, 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, 9, 0, 1, 10, 21, 14, 0, 1, 9, 27, 16, 1, 0, 1, 10, 29, 22, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 COMMENTS The boundary size is the number of parts having fewer than two neighbors. LINKS Alois P. Heinz, Rows n = 0..600, flattened EXAMPLE T(12,1) = 1: [12]. T(12,2) = 6: [1,11], [2,10], [3,4,5], [3,9], [4,8], [5,7]. 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]. T(12,4) = 1: [1,2,4,5]. Triangle T(n,k) begins:   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, 1, 5, 6;   0, 1, 6, 7, 1; MAPLE b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),       expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+       `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))     end: T:= n-> (p->seq(coeff(p, x, i), i=0..degree(p)))(b(n\$2, 0)): seq(T(n), n=0..30); MATHEMATICA 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 *) CROSSREFS Columns k=0-10 give: A000007, A057427, A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567. Row sums give: A000009. Last elements of rows give: A227552. Cf. A227345 (a version with trailing zeros), A053993, A201077, A227568, A224878 (one part of size 0 allowed). Sequence in context: A308999 A208343 A029324 * A029318 A210381 A029297 Adjacent sequences:  A227548 A227549 A227550 * A227552 A227553 A227554 KEYWORD nonn,look,tabf AUTHOR Alois P. Heinz, Jul 16 2013 STATUS approved

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Last modified September 19 11:06 EDT 2019. Contains 327192 sequences. (Running on oeis4.)