A029869 -id:A029869 - cp's OEIS frontend

A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 6, 3, 1, 1, 20, 16, 9, 4, 1, 1, 48, 37, 23, 11, 4, 1, 1, 115, 96, 62, 35, 14, 5, 1, 1, 286, 239, 169, 97, 46, 18, 5, 1, 1, 719, 622, 451, 282, 145, 63, 21, 6, 1, 1, 1842, 1607, 1217, 792, 440, 206, 80, 25, 6, 1, 1

Offset: 1

Andrew Howroyd, Dec 03 2020

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    4,   3,   1,   1;
    9,   6,   3,   1,   1;
   20,  16,   9,   4,   1,  1;
   48,  37,  23,  11,   4,  1,  1;
  115,  96,  62,  35,  14,  5,  1, 1;
  286, 239, 169,  97,  46, 18,  5, 1, 1;
  719, 622, 451, 282, 145, 63, 21, 6, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Columns 1..12 are A000081, A027852, A029852, A029853, A029868, A029869, A029870, A029871, A032205, A032206, A032207, A032208.
Row sums are A002861.
Cf. A033185, A217781, A339067.

\\ TreeGf is A000081 as g.f.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
ColSeq(n,k)={my(r=TreeGf(max(0,n+1-k))); Vec(sumdiv(k, d, eulerphi(d)*subst(r + O(x*x^(n\d)), x, x^d)^(k/d))/k, -n)}
M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

G.f. of k-th column: (1/k)*Sum_{d|k} phi(d) * r(x^d)^(k/d) where r(x) is the g.f. of A000081.

A241188 Triangle T(n,s) of Dynkin type D_n read by rows (n >= 2, 0 <= s <= n).

Original entry on oeis.org

1, 2, 1, 1, 3, 5, 5, 1, 4, 9, 16, 20, 1, 5, 14, 30, 55, 77, 1, 6, 20, 50, 105, 196, 294, 1, 7, 27, 77, 182, 378, 714, 1122, 1, 8, 35, 112, 294, 672, 1386, 2640, 4290, 1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445

Offset: 2

N. J. A. Sloane, Apr 24 2014

			Triangle begins:
1, 2, 1,
1, 3, 5, 5,
1, 4, 9, 16, 20,
1, 5, 14, 30, 55, 77,
1, 6, 20, 50, 105, 196, 294,
1, 7, 27, 77, 182, 378, 714, 1122,
1, 8, 35, 112, 294, 672, 1386, 2640, 4290,
1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 16445,
...

M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014.
M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, arXiv:1403.5827 [math.RT], 2014 and J. Int. Seq. 18 (2015) 15.10.6.

See A009766 for the case of type A.
See A059481 for the case of type B/C.
Diagonals give A029869, A051960, A029651, A051924. Row sums are also A051924.

f[t_, s_] := Binomial[t, s] (s + t)/t;
T[, 0] = 1; T[n, n_] := f[2 n - 2, n - 2]; T[n_, s_] := f[n + s - 2, s];
Table[T[n, s], {n, 2, 9}, {s, 0, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

T(n,s) = [n+s-2,s] for 0 <= s < n, T(n,n) = [2n-2,n-2], where [t,s] stands for binomial(t,s)*(s+t)/t.

cp's OEIS Frontend

A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula