A238342 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the smallest part, n>=0, 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 3, 4, 0, 1, 0, 8, 3, 4, 0, 1, 0, 11, 10, 5, 5, 0, 1, 0, 20, 18, 14, 5, 6, 0, 1, 0, 34, 35, 24, 21, 6, 7, 0, 1, 0, 59, 60, 59, 35, 27, 7, 8, 0, 1, 0, 96, 121, 108, 85, 49, 35, 8, 9, 0, 1, 0, 167, 217, 213, 175, 125, 63, 44, 9, 10, 0, 1, 0, 282, 391, 419, 366, 258, 176, 80, 54, 10, 11, 0, 1

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Keywords

• nonn
• tabl

Comments

Conjecture: Generally, for k > 0 is a(n) ~ n^k * ((1+sqrt(5))/2)^(n-2*k-1) / (5^((k+1)/2) * k!). Holds for all k<=10. - Vaclav Kotesovec, May 02 2014

G.f.: 1 + Sum_{i>0} (-y*(x^i)*(x - 1)^2)/( (x^(i+1) + x - 1)*((x^i)*(x*(y - 1) - y) - x + 1) ). - John Tyler Rascoe, Oct 15 2024

Sum_{k=0..n} k * T(n,k) = A097941(n). - Alois P. Heinz, Oct 15 2024

Examples

			Triangle starts:
00:  1;
01:  0,    1;
02:  0,    1,    1;
03:  0,    3,    0,    1;
04:  0,    3,    4,    0,    1;
05:  0,    8,    3,    4,    0,    1;
06:  0,   11,   10,    5,    5,    0,    1;
07:  0,   20,   18,   14,    5,    6,    0,    1;
08:  0,   34,   35,   24,   21,    6,    7,    0,   1;
09:  0,   59,   60,   59,   35,   27,    7,    8,   0,   1;
10:  0,   96,  121,  108,   85,   49,   35,    8,   9,   0,   1;
11:  0,  167,  217,  213,  175,  125,   63,   44,   9,  10,   0,  1;
12:  0,  282,  391,  419,  366,  258,  176,   80,  54,  10,  11,  0,  1;
13:  0,  475,  709,  808,  730,  579,  371,  236,  99,  65,  11, 12,  0,  1;
14:  0,  800, 1281, 1522, 1481, 1202,  861,  513, 309, 120,  77, 12, 13,  0, 1;
15:  0, 1352, 2283, 2872, 2925, 2512, 1862, 1238, 684, 395, 143, 90, 13, 14, 0, 1;
...
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A238341 (the same for largest part).

Columns k=0-10 give: A000007, A105039, A241862, A241863, A241864, A241865, A241866, A241867, A241868, A241869, A241870.

Row sums are A011782.

T(2*n,n) gives A232665(n).

Cf. A097941.

Programs

Maple

b:= proc(n, s) option remember;`if`(n=0, 1,
      `if`(n`if`(k=0, `if`(n=0, 1, 0), add((p->add(coeff(p, x, i)*
     binomial(i+k, k), i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)):
seq(seq(T(n, k), k=0..n), n=0..15);

Mathematica

b[n_, s_] := b[n, s] = If[n == 0, 1, If[nJean-François Alcover, Nov 07 2014, translated from Maple *)

PARI

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h=1+sum(i=1,N,(-y*(x^i)*(x-1)^2)/((x^(i+1)+x-1)*((x^i)*(x*(y-1)-y)-x+1)))); for(i=0,N-1, print(Vecrev(polcoef(h,i))))}
T_xy(15) \\ John Tyler Rascoe, Oct 15 2024