cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A020555 Number of multigraphs on n labeled edges (with loops). Also number of genetically distinct states amongst n individuals.

Original entry on oeis.org

1, 2, 9, 66, 712, 10457, 198091, 4659138, 132315780, 4441561814, 173290498279, 7751828612725, 393110572846777, 22385579339430539, 1419799938299929267, 99593312799819072788, 7678949893962472351181, 647265784993486603555551, 59357523410046023899154274
Offset: 0

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Author

Gilbert Labelle (gilbert(AT)lacim.uqam.ca), Simon Plouffe, N. J. A. Sloane

Keywords

Comments

Also the number of factorizations of (p_n#)^2. - David W. Wilson, Apr 30 2001
Also the number of multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018
a(n) gives the number of genetically distinct states for n diploid individuals in the case that maternal and paternal alleles transmitted to the individuals are not distinguished (if maternal and paternal alleles are distinguished, then the number of states is A000110(2n)). - Noah A Rosenberg, Aug 23 2022

Examples

			From _Gus Wiseman_, Jul 18 2018: (Start)
The a(2) = 9 multiset partitions of {1, 1, 2, 2}:
  (1122),
  (1)(122), (2)(112), (11)(22), (12)(12),
  (1)(1)(22), (1)(2)(12), (2)(2)(11),
  (1)(1)(2)(2).
(End)

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
  • E. Keith Lloyd, Math. Proc. Camb. Phil. Soc., vol. 103 (1988), 277-284.
  • A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
  • G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Links

Crossrefs

Row n=2 of A219727. - Alois P. Heinz, Nov 26 2012
Cf. A007716, A007717, A014500, A014501, A020554, A094574, A316974.
See also A322764. Row 0 of the array in A322765.
Main diagonal of A346500.
Cf. A001055, A002110.

Programs

  • Maple
    B := n -> combinat[bell](n):
P := proc(m,n) local k; global B; option remember;
if n = 0 then B(m)  else
(1/2)*( P(m+2,n-1) + P(m+1,n-1) + add( binomial(n-1,k)*P(m,k), k=0..n-1) ); fi; end;
r:=m->[seq(P(m,n),n=0..20)]; r(0); # N. J. A. Sloane, Dec 30 2018
  • Mathematica
    max = 16; s = Series[Exp[-3/2 + Exp[x]/2]*Sum[Exp[Binomial[n+1, 2]*x]/n!, {n, 0, 3*max }], {x, 0, max}] // Normal; a[n_] := SeriesCoefficient[s, {x, 0, n}]*n!; Table[a[n] // Round, {n, 0, max} ] (* Jean-François Alcover, Apr 23 2014, after Vladeta Jovovic *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[mps[Ceiling[Range[1/2,n,1/2]]]],{n,5}] (* Gus Wiseman, Jul 18 2018 *)

Formula

Lloyd's article gives a complicated explicit formula.
E.g.f.: exp(-3/2 + exp(x)/2)*Sum_{n>=0} exp(binomial(n+1, 2)*x)/n! [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004
a(n) = A001055(A002110(n)^2). - Alois P. Heinz, Aug 23 2022

A346500 Number A(n,k) of partitions of the (n+k)-multiset {1,2,...,n,1,2,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 4, 4, 5, 15, 11, 9, 11, 15, 52, 36, 26, 26, 36, 52, 203, 135, 92, 66, 92, 135, 203, 877, 566, 371, 249, 249, 371, 566, 877, 4140, 2610, 1663, 1075, 712, 1075, 1663, 2610, 4140, 21147, 13082, 8155, 5133, 3274, 3274, 5133, 8155, 13082, 21147
Offset: 0

Views

Author

Alois P. Heinz, Jul 20 2021

Keywords

Comments

Also number A(n,k) of factorizations of Product_{i=1..n} prime(i) * Product_{i=1..k} prime(i); A(2,2) = 9: 2*2*3*3, 3*3*4, 6*6, 2*3*6, 4*9, 2*2*9, 3*12, 2*18, 36.

Examples

			A(2,2) = 9: 1122, 11|22, 12|12, 1|122, 112|2, 11|2|2, 1|1|22, 1|12|2, 1|1|2|2.
Square array A(n,k) begins:
    1,    1,    2,     5,    15,     52,     203,     877, ...
    1,    2,    4,    11,    36,    135,     566,    2610, ...
    2,    4,    9,    26,    92,    371,    1663,    8155, ...
    5,   11,   26,    66,   249,   1075,    5133,   26683, ...
   15,   36,   92,   249,   712,   3274,   16601,   91226, ...
   52,  135,  371,  1075,  3274,  10457,   56135,  325269, ...
  203,  566, 1663,  5133, 16601,  56135,  198091, 1207433, ...
  877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, ...
  ...

Links

Crossrefs

Columns (or rows) k=0-10 give: A000110, A035098, A322764, A322768, A346881, A346882, A346883, A346884, A346885, A346886, A346887.
Main diagonal gives A020555.
First upper (or lower) diagonal gives A322766.
Second upper (or lower) diagonal gives A322767.
Antidiagonal sums give A346490.
A(2n,n) gives A322769.
Cf. A001055, A002110, A322765, A346426, A346517.

Programs

  • Maple
    g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
     `if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
        g(n/d, d)), d=divisors(n) minus {1, n}))
    end:
p:= proc(n) option remember; `if`(n=0, 1, p(n-1)*ithprime(n)) end:
A:= (n, k)-> g(p(n)*p(k)$2):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
A:= proc(n, k) option remember; `if`(n
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
A[n_, k_] := A[n, k] = If[n < k, A[k, n],
     If[k == 0, b[n], (A[n + 1, k - 1] + Sum[A[n - k + j, j]*
     Binomial[k - 1, j], {j, 0, k - 1}] + A[n, k - 1])/2]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz's second program *)

Formula

A(n,k) = A001055(A002110(n)*A002110(k)).
A(n,k) = A(k,n).
A(n,k) = A322765(abs(n-k),min(n,k)).

A322765 Array read by upwards antidiagonals: T(m,n) = number of set partitions of the multiset consisting of one copy each of x_1, x_2, ..., x_m, and two copies each of y_1, y_2, ..., y_n, for m >= 0, n >= 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 9, 5, 11, 26, 66, 15, 36, 92, 249, 712, 52, 135, 371, 1075, 3274, 10457, 203, 566, 1663, 5133, 16601, 56135, 198091, 877, 2610, 8155, 26683, 91226, 325269, 1207433, 4659138, 4140, 13082, 43263, 149410, 537813, 2014321, 7837862, 31638625, 132315780
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Examples

			The array begins:
    1,    2,     9,     66,      712,     10457,      198091, ...
    1,    4,    26,    249,     3274,     56135,     1207433, ...
    2,   11,    92,   1075,    16601,    325269,     7837862, ...
    5,   36,   371,   5133,    91226,   2014321,    53840640, ...
   15,  135,  1663,  26683,   537813,  13241402,   389498179, ...
   52,  566,  8155, 149410,  3376696,  91914202,  2955909119, ...
  203, 2610, 43263, 894124, 22451030, 670867539, 23456071495, ...
  ...

References

  • D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778.

Links

Crossrefs

Rows include A020555, A322766, A322767.
Columns include A000110, A035098, A322764, A322768.
Main diagonal is A322769.
See A322770 for partitions into distinct parts.
Cf. A001055, A002110, A346500.

Programs

  • Maple
    B := n -> combinat[bell](n):
P := proc(m,n) local k; global B; option remember;
if n = 0 then B(m)  else
(1/2)*( P(m+2,n-1) + P(m+1,n-1) + add( binomial(n-1,k)*P(m,k), k=0..n-1) ); fi; end; # P(m,n) (which is Knuth's notation) is T(m,n)
  • Mathematica
    P[m_, n_] := P[m, n] = If[n == 0, BellB[m], (1/2)(P[m+2, n-1] + P[m+1, n-1] + Sum[Binomial[n-1, k] P[m, k], {k, 0, n-1}])];
Table[P[m-n, n], {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 02 2019, from Maple *)
  • PARI
    {T(n, k) = if(k==0, sum(j=0, n, stirling(n, j, 2)), (T(n+2, k-1)+T(n+1, k-1)+sum(j=0, k-1, binomial(k-1, j)*T(n, j)))/2)} \\ Seiichi Manyama, Nov 21 2020

Formula

Knuth p. 779 gives a recurrence using the Bell numbers A000110 (see Maple program).
From Alois P. Heinz, Jul 21 2021: (Start)
A(n,k) = A001055(A002110(n+k)*A002110(k)).
A(n,k) = A346500(n+k,k). (End)

A322773 Column 2 of array in A322770.

Original entry on oeis.org

5, 18, 70, 299, 1393, 7023, 38043, 220054, 1352082, 8784991, 60125371, 432001747, 3248914361, 25508188118, 208592396802, 1772921926183, 15632838989393, 142759592985079, 1348095912827295, 13145321614286610, 132188675368994446, 1369216940917868547
Offset: 0

Views

Author

N. J. A. Sloane, Dec 30 2018

Keywords

Links

Crossrefs

Cf. A322764, A322770.
Column k=2 of A346517.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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