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A135429 Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 3 for all x. 3
1, 1, 4, 29, 210, 2116, 25522, 362832, 6000276, 113593688, 2434603356, 58523364604, 1565365441708, 46273309903536, 1502773485741816, 53336787604185656, 2059209704215556448, 86117458019804680576, 3886421648246467359364, 188615552477984650605744 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..150

Alois P. Heinz, Illustration with formula for a(n)

FORMULA

a(n) = see program.

MAPLE

A025035:= proc(n) option remember; (3*n)! /n! /(6^n) end:

z:= proc(n) option remember; add(binomial(n, k+k) *doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2)) end:

r:= proc(n) option remember; n! * add(add(add(add(Stirling2(e, d) *a^(d+i) *(a*(a+1)/2)^(n-i-i-e-d-a) /a! /(n-i-i-e-d-a)! /i! /e! /(2^i), a=0..(n-i-i-e-d)), d=0..min(e, n-i-i-e)), e=0..(n-i-i)), i=0..floor(n/2)) end:

a:= proc(n) option remember; n! *add(add(A025035(i) *z(j) *r(n-3*i-j) /(3*i)! /j! /(n-3*i-j)!, j=0..(n-3*i)), i=0..floor(n/3)) end:

seq(a(n), n=0..30);

MATHEMATICA

Unprotect[Power]; 0^0 = 1; A025035[n_] := A025035[n] = (3n)!/n!/6^n; z[n_] := z[n] = Sum[Binomial[n, k+k]*(k+k-1)!!*k^(n-k-k), {k, 0, Floor[n/2]}]; r[n_] := r[n] = n!*Sum[Sum[Sum[Sum[StirlingS2[e, d]*a^(d+i)*(a*(a+1)/2)^(n-i-i-e-d-a)/a!/(n-i-i-e-d-a)!/i!/e!/2^i, {a, 0, n-i-i-e-d}], {d, 0, Min[e, n-i-i-e]}], {e, 0, n-i-i}], {i, 0, Floor[n/2]}]; a[n_] := a[n] = n!*Sum[Sum[A025035[i]*z[j]*r[n-3*i-j]/(3i)!/j!/(n-3*i-j)!, {j, 0, n-3*i}], {i, 0, Floor[n/3]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A135312, A025035, A006882, A008277, A000142.

Sequence in context: A100022 A001883 A281600 * A079756 A344098 A221415

Adjacent sequences:  A135426 A135427 A135428 * A135430 A135431 A135432

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Dec 12 2007

STATUS

approved

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Last modified October 25 11:00 EDT 2021. Contains 348239 sequences. (Running on oeis4.)