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A217859 Triangular array read by rows. T(n,k) is the number of functions on n unlabeled nodes that have exactly k unique components (n >= 1, k >= 1). 0
1, 3, 5, 2, 12, 7, 21, 25, 1, 58, 63, 9, 126, 178, 39, 341, 466, 140, 4, 867, 1253, 470, 25, 2334, 3418, 1431, 135, 6218, 9365, 4358, 544, 6, 17016, 25924, 12871, 2042, 50, 46351, 72207, 37993, 7056, 291, 127842, 202345, 111142, 23483, 1383, 4, 353297, 568822, 325359, 75701, 5754, 60 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums are A001372.

T(n,1) = A002861(n) + 1 when n is prime (counts connected functions and the identity function).

LINKS

Table of n, a(n) for n=1..54.

N. J. A. Sloane, Illustration of initial terms

FORMULA

O.g.f.: Product_{n>=1} ((y*x^n - x^n + 1)/(1 - x^n))^A002861(n).

EXAMPLE

Triangle begins:

       1;

       3,

       5,      2;

      12,      7;

      21,     25,      1;

      58,     63,      9;

     126,    178,     39;

     341,    466,    140,     4;

     867,   1253,    470,    25;

    2334,   3418,   1431,   135;

    6218,   9365,   4358,   544,    6;

   17016,  25924,  12871,  2042,   50;

   46351,  72207,  37993,  7056,  291;

  127842, 202345, 111142, 23483, 1383,  4;

  353297, 568822, 325359, 75701, 5754, 60;

T(3,2)=2 because (in the link) the third and the fifth digraphs on 3 nodes are composed of 2 unique components.

MATHEMATICA

Needs["Combinatorica`"];

nn=30; s[n_, k_]:=s[n, k]=a[n+1-k]+If[n<2 k, 0, s[n-k, k]]; a[1]=1; a[n_]:=a[n]=Sum[a[i] s[n-1, i] i, {i, 1, n-1}]/(n-1); rt=Table[a[i], {i, 1, nn}]; c=Drop[Apply[Plus, Table[Take[CoefficientList[CycleIndex[CyclicGroup[n], s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i), {i, 1, nn}], {k, 1, nn}][[j]], {j, 1, nn}], x], nn], {n, 1, 30}]], 1]; CoefficientList[Series[Product[((y x^i +1-x^i)/(1-x^i))^c[[i]], {i, 1, nn-1}], {x, 0, 15}], {x, y}]//Grid

(* after code given by Robert A. Russell in A000081 *)

CROSSREFS

Sequence in context: A076556 A324776 A325891 * A108426 A301305 A163237

Adjacent sequences:  A217856 A217857 A217858 * A217860 A217861 A217862

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Oct 13 2012

STATUS

approved

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Last modified February 26 16:54 EST 2021. Contains 341632 sequences. (Running on oeis4.)