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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 (list; table; graph; refs; listen; history; text; internal format)
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).

LINKS

Joerg Arndt, Table of n, a(n) for n = 1..5050

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-Fran├žois 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: A263764 A070202 A280129 * A130207 A167688 A083914

Adjacent sequences:  A227341 A227342 A227343 * A227345 A227346 A227347

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt, Jul 08 2013

STATUS

approved

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Last modified November 23 19:03 EST 2017. Contains 295128 sequences.