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%I #28 Mar 06 2016 16:40:53
%S 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,
%T 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,
%U 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
%N 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.
%C 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
%H Joerg Arndt and Alois P. Heinz, <a href="/A238342/b238342.txt">Rows n = 0..140, flattened</a>
%e Triangle starts:
%e 00: 1;
%e 01: 0, 1;
%e 02: 0, 1, 1;
%e 03: 0, 3, 0, 1;
%e 04: 0, 3, 4, 0, 1;
%e 05: 0, 8, 3, 4, 0, 1;
%e 06: 0, 11, 10, 5, 5, 0, 1;
%e 07: 0, 20, 18, 14, 5, 6, 0, 1;
%e 08: 0, 34, 35, 24, 21, 6, 7, 0, 1;
%e 09: 0, 59, 60, 59, 35, 27, 7, 8, 0, 1;
%e 10: 0, 96, 121, 108, 85, 49, 35, 8, 9, 0, 1;
%e 11: 0, 167, 217, 213, 175, 125, 63, 44, 9, 10, 0, 1;
%e 12: 0, 282, 391, 419, 366, 258, 176, 80, 54, 10, 11, 0, 1;
%e 13: 0, 475, 709, 808, 730, 579, 371, 236, 99, 65, 11, 12, 0, 1;
%e 14: 0, 800, 1281, 1522, 1481, 1202, 861, 513, 309, 120, 77, 12, 13, 0, 1;
%e 15: 0, 1352, 2283, 2872, 2925, 2512, 1862, 1238, 684, 395, 143, 90, 13, 14, 0, 1;
%e ...
%p b:= proc(n, s) option remember;`if`(n=0, 1,
%p `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
%p end:
%p T:= (n, k)->`if`(k=0, `if`(n=0, 1, 0), add((p->add(coeff(p, x, i)*
%p binomial(i+k, k), i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)):
%p seq(seq(T(n, k), k=0..n), n=0..15);
%t b[n_, s_] := b[n, s] = If[n == 0, 1, If[n<s, 0, Expand[Sum[b[n-j, s]*x, {j, s, n}]]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i+k, k], {i, 0, Exponent[p, x]}]][b[n-j*k, j+1]], {j, 1, n/k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Nov 07 2014, translated from Maple *)
%Y Cf. A238341 (the same for largest part).
%Y Columns k=0-10 give: A000007, A105039, A241862, A241863, A241864, A241865, A241866, A241867, A241868, A241869, A241870.
%Y Row sums are A011782.
%Y T(2*n,n) gives A232665(n).
%K nonn,tabl
%O 0,8
%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014
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