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%I A238342 #32 Oct 15 2024 15:48:21
%S A238342 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 A238342 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 A238342 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 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.
%C A238342 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
%C A238342 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
%C A238342 Sum_{k=0..n} k * T(n,k) = A097941(n). - _Alois P. Heinz_, Oct 15 2024
%H A238342 Joerg Arndt and Alois P. Heinz, <a href="/A238342/b238342.txt">Rows n = 0..140, flattened</a>
%e A238342 Triangle starts:
%e A238342 00:  1;
%e A238342 01:  0,    1;
%e A238342 02:  0,    1,    1;
%e A238342 03:  0,    3,    0,    1;
%e A238342 04:  0,    3,    4,    0,    1;
%e A238342 05:  0,    8,    3,    4,    0,    1;
%e A238342 06:  0,   11,   10,    5,    5,    0,    1;
%e A238342 07:  0,   20,   18,   14,    5,    6,    0,    1;
%e A238342 08:  0,   34,   35,   24,   21,    6,    7,    0,   1;
%e A238342 09:  0,   59,   60,   59,   35,   27,    7,    8,   0,   1;
%e A238342 10:  0,   96,  121,  108,   85,   49,   35,    8,   9,   0,   1;
%e A238342 11:  0,  167,  217,  213,  175,  125,   63,   44,   9,  10,   0,  1;
%e A238342 12:  0,  282,  391,  419,  366,  258,  176,   80,  54,  10,  11,  0,  1;
%e A238342 13:  0,  475,  709,  808,  730,  579,  371,  236,  99,  65,  11, 12,  0,  1;
%e A238342 14:  0,  800, 1281, 1522, 1481, 1202,  861,  513, 309, 120,  77, 12, 13,  0, 1;
%e A238342 15:  0, 1352, 2283, 2872, 2925, 2512, 1862, 1238, 684, 395, 143, 90, 13, 14, 0, 1;
%e A238342 ...
%p A238342 b:= proc(n, s) option remember;`if`(n=0, 1,
%p A238342       `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
%p A238342     end:
%p A238342 T:= (n, k)->`if`(k=0, `if`(n=0, 1, 0), add((p->add(coeff(p, x, i)*
%p A238342      binomial(i+k, k), i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)):
%p A238342 seq(seq(T(n, k), k=0..n), n=0..15);
%t A238342 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 *)
%o A238342 (PARI)
%o A238342 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))))}
%o A238342 T_xy(15) \\ _John Tyler Rascoe_, Oct 15 2024
%Y A238342 Cf. A238341 (the same for largest part).
%Y A238342 Columns k=0-10 give: A000007, A105039, A241862, A241863, A241864, A241865, A241866, A241867, A241868, A241869, A241870.
%Y A238342 Row sums are A011782.
%Y A238342 T(2*n,n) gives A232665(n).
%Y A238342 Cf. A097941.
%K A238342 nonn,tabl
%O A238342 0,8
%A A238342 _Joerg Arndt_ and _Alois P. Heinz_, Feb 25 2014

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