|
%I #21 Jan 28 2015 04:15:03
%S 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,
%T 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,
%U 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
%N Triangle read by rows, partitions into distinct parts by perimeter.
%C 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).
%C Row sums are A000009.
%C Column sums are A122129 (noted by _Patrick Devlin_).
%H Joerg Arndt, <a href="/A227344/b227344.txt">Table of n, a(n) for n = 1..5050</a>
%e Triangle starts (dots for zeros):
%e 01: 1
%e 02: . 1
%e 03: . . 2
%e 04: . . . 2
%e 05: . . . . 3
%e 06: . . . 1 . 3
%e 07: . . . . . . 5
%e 08: . . . . . . . 6
%e 09: . . . . . 1 . . 7
%e 10: . . . . 1 . . . . 9
%e 11: . . . . . . . . 1 . 11
%e 12: . . . . . . . 1 . 1 . 13
%e 13: . . . . . . . . 1 . 1 . 16
%e 14: . . . . . . 1 . . . . 1 . 20
%e 15: . . . . . 1 . . . 1 . 1 1 . 23
%e 16: . . . . . . . . . . 2 . 1 1 . 28
%e 17: . . . . . . . . . . . 2 . 1 2 . 33
%e 18: . . . . . . . . 1 . . 1 2 . 1 2 . 39
%e 19: . . . . . . . . . 1 . . 1 1 1 1 3 . 46
%e 20: . . . . . . . 1 . . . . . 1 1 2 1 3 . 55
%e 21: . . . . . . 1 . . . . . . 2 2 1 2 1 4 . 63
%e 22: . . . . . . . . . . 1 . 1 . 2 1 1 2 2 4 . 75
%e 23: . . . . . . . . . . . 1 . 1 . 2 1 3 2 2 5 . 87
%e 24: . . . . . . . . . . . . 1 . 1 2 3 . 4 2 3 5 . 101
%e 25: . . . . . . . . . 1 . . . 1 . 1 1 3 . 6 2 3 7 . 117
%e 26: . . . . . . . . . . 1 . 1 . . . 2 1 3 . 7 2 4 8 . 136
%e 27: . . . . . . . . 1 . . . . 1 . . . 5 2 2 1 8 3 4 9 . 156
%e 28: . . . . . . . 1 . . . . . . 1 1 . . 4 2 3 2 8 4 5 11 . 180
%e 29: . . . . . . . . . . . . . . 1 2 1 . . 4 3 3 3 9 5 5 13 . 207
%e 30: . . . . . . . . . . . 1 . . 1 1 1 1 . 3 6 2 2 5 9 6 6 14 . 238
%p b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x^(i+1), 1),
%p expand(`if`(i<1, 0, `if`(t>1, x^(i+1), 1)*b(n, i-1, iquo(t, 2))+
%p `if`(i>n, 0, `if`(t=2, x^(i+1), 1)*b(n-i, i-1, iquo(t, 2)+2)))))
%p end:
%p T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
%p seq(T(n), n=1..20); # _Alois P. Heinz_, Jul 16 2013
%t 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_ *)
%Y Cf. A227345 (partitions by boundary size).
%Y Cf. A227426 (diagonal: number of partitions with maximal perimeter).
%Y Cf. A227538 (smallest k with positive T(n,k)), A227614 (second lower diagonal). - _Alois P. Heinz_, Jul 17 2013
%K nonn,tabl
%O 1,6
%A _Joerg Arndt_, Jul 08 2013
|
