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A227344
Triangle read by rows, partitions into distinct parts by perimeter.
5
1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 1, 0, 0, 7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 16, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 20, 0
OFFSET
1,6
COMMENTS
The perimeter of a partition is the sum of all parts p that do not have two neighbors (that is, not both p-1 and p+1 are parts).
Row sums are A000009.
Column sums are A122129 (noted by Patrick Devlin).
EXAMPLE
Triangle starts (dots for zeros):
01: 1
02: . 1
03: . . 2
04: . . . 2
05: . . . . 3
06: . . . 1 . 3
07: . . . . . . 5
08: . . . . . . . 6
09: . . . . . 1 . . 7
10: . . . . 1 . . . . 9
11: . . . . . . . . 1 . 11
12: . . . . . . . 1 . 1 . 13
13: . . . . . . . . 1 . 1 . 16
14: . . . . . . 1 . . . . 1 . 20
15: . . . . . 1 . . . 1 . 1 1 . 23
16: . . . . . . . . . . 2 . 1 1 . 28
17: . . . . . . . . . . . 2 . 1 2 . 33
18: . . . . . . . . 1 . . 1 2 . 1 2 . 39
19: . . . . . . . . . 1 . . 1 1 1 1 3 . 46
20: . . . . . . . 1 . . . . . 1 1 2 1 3 . 55
21: . . . . . . 1 . . . . . . 2 2 1 2 1 4 . 63
22: . . . . . . . . . . 1 . 1 . 2 1 1 2 2 4 . 75
23: . . . . . . . . . . . 1 . 1 . 2 1 3 2 2 5 . 87
24: . . . . . . . . . . . . 1 . 1 2 3 . 4 2 3 5 . 101
25: . . . . . . . . . 1 . . . 1 . 1 1 3 . 6 2 3 7 . 117
26: . . . . . . . . . . 1 . 1 . . . 2 1 3 . 7 2 4 8 . 136
27: . . . . . . . . 1 . . . . 1 . . . 5 2 2 1 8 3 4 9 . 156
28: . . . . . . . 1 . . . . . . 1 1 . . 4 2 3 2 8 4 5 11 . 180
29: . . . . . . . . . . . . . . 1 2 1 . . 4 3 3 3 9 5 5 13 . 207
30: . . . . . . . . . . . 1 . . 1 1 1 1 . 3 6 2 2 5 9 6 6 14 . 238
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x^(i+1), 1),
expand(`if`(i<1, 0, `if`(t>1, x^(i+1), 1)*b(n, i-1, iquo(t, 2))+
`if`(i>n, 0, `if`(t=2, x^(i+1), 1)*b(n-i, i-1, iquo(t, 2)+2)))))
end:
T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
seq(T(n), n=1..20); # Alois P. Heinz, Jul 16 2013
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t>1, x^(i+1), 1], Expand[If[i<1, 0, If[t>1, x^(i+1), 1]*b[n, i-1, Quotient[t, 2]] + If[i>n, 0, If[t == 2, x^(i+1), 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, Jan 28 2015, after Alois P. Heinz *)
CROSSREFS
Cf. A227345 (partitions by boundary size).
Cf. A227426 (diagonal: number of partitions with maximal perimeter).
Cf. A227538 (smallest k with positive T(n,k)), A227614 (second lower diagonal). - Alois P. Heinz, Jul 17 2013
Sequence in context: A325668 A070202 A280129 * A130207 A325433 A167688
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, Jul 08 2013
STATUS
approved