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.

A061260 G.f.: Product_{k>=1} (1-y*x^k)^(-numbpart(k)), where numbpart(k) = number of partitions of k, cf. A000041.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 6, 2, 1, 7, 11, 6, 2, 1, 11, 23, 15, 6, 2, 1, 15, 40, 32, 15, 6, 2, 1, 22, 73, 67, 37, 15, 6, 2, 1, 30, 120, 134, 79, 37, 15, 6, 2, 1, 42, 202, 255, 172, 85, 37, 15, 6, 2, 1, 56, 320, 470, 348, 187, 85, 37, 15, 6, 2, 1, 77, 511, 848, 697, 397, 194, 85, 37, 15, 6, 2, 1
Offset: 1

Views

Author

Vladeta Jovovic, Apr 23 2001

Keywords

Comments

Multiset transformation of A000041. - R. J. Mathar, Apr 30 2017
Number of orderless twice-partitions of n of length k. A twice-partition of n is a choice of a partition of each part in a partition of n. The T(5,3) = 6 orderless twice-partitions: (3)(1)(1), (21)(1)(1), (111)(1)(1), (2)(2)(1), (2)(11)(1), (11)(11)(1). - Gus Wiseman, Mar 23 2018

Examples

			:  1;
:  2,   1;
:  3,   2,   1;
:  5,   6,   2,   1;
:  7,  11,   6,   2,  1;
: 11,  23,  15,   6,  2,  1;
: 15,  40,  32,  15,  6,  2,  1;
: 22,  73,  67,  37, 15,  6,  2, 1;
: 30, 120, 134,  79, 37, 15,  6, 2, 1;
: 42, 202, 255, 172, 85, 37, 15, 6, 2, 1;

Links

Crossrefs

Row sums: A001970, first column: A000041.
T(2,n) gives A061261,
Cf. A063834, A119442, A273873, A285229, A289078, A289501, A299200, A299201.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*binomial(
       combinat[numbpart](i)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[PartitionsP[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 17 2018, after Alois P. Heinz *)

A301706 Number of rooted thrice-partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 91, 201, 422, 918, 1896, 4089, 8376, 17793, 36445, 76446, 155209, 324481, 655426, 1355220, 2741092, 5617505, 11291037, 23086423, 46227338, 93753196, 187754647, 378675055, 754695631, 1518414812, 3016719277, 6037006608, 11984729983
Offset: 1

Views

Author

Gus Wiseman, Mar 25 2018

Keywords

Comments

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.

Examples

			The a(5) = 9 rooted thrice-partitions:
((2)), ((11)), ((1)()), (()()()),
((1))(), (()())(), (())(()),
(())()(),
()()()().
The a(6) = 19 rooted thrice-partitions:
((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()),
((2))(), ((11))(), ((1)())(), (()()())(), ((1))(()), (()())(()),
((1))()(), (()())()(), (())(())(),
(())()()(),
()()()()().

Crossrefs

Cf. A000041, A001383, A002865, A063834, A093637, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301467, A301480, A301595, A301598.

Programs

  • Mathematica
    twire[n_]:=twire[n]=Sum[Times@@PartitionsP/@(ptn-1),{ptn,IntegerPartitions[n-1]}];
thrire[n_]:=Sum[Times@@twire/@ptn,{ptn,IntegerPartitions[n-1]}];
Array[thrire,30]

A119441 Distribution of A063834 in Abramowitz and Stegun order.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 11, 7, 10, 9, 5, 6, 8, 3, 4, 2, 1, 15, 11, 14, 15, 7, 10, 9, 12, 5, 6, 8, 3, 4, 2, 1, 22, 15, 22, 21, 25, 11, 14, 15, 20, 18, 7, 10, 9, 12, 16, 5, 6, 8, 3, 4, 2, 1, 30, 22, 30, 33, 35, 15, 22, 21
Offset: 1

Views

Author

Alford Arnold, May 19 2006

Keywords

Examples

			1;
2, 1;
3, 2, 1;
5, 3, 4, 2, 1;
7, 5, 6, 3, 4, 2, 1;
T(5,2) = 5 because the second partition of 5 is 1+4 and 4 can be repartitioned in 5 different ways.
T(5,3) = 6 because the third partition of 5 is 2+3, where the 2 can be partitioned in 2 ways (2, 1+1) and the 3 can be partitioned in 3 ways (3, 1+2, 1+1+1), 6=2*3.
T(5,4) = 3 because the fourth partition of 5 is 1+1+3 and 3 can be partitioned in 3 different ways.

Links

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A063834, A119442, A000041 (row lengths and also first column)

Programs

  • Maple
    # Compare two partitions (list) in AS order.
AScompare := proc(p1,p2)
    if nops(p1) > nops(p2) then
        return 1;
    elif nops(p1) < nops(p2) then
        return -1;
    else
        for i from 1 to nops(p1) do
            if op(i,p1) > op(i,p2) then
                return 1;
            elif op(i,p1) < op(i,p2) then
                return -1;
            end if;
        end do:
        return 0 ;
    end if;
end proc:
# Produce list of partitions in AS order
ASPrts := proc(n)
    local pi,insrt,p,ex ;
    pi := [] ;
    for p in combinat[partition](n) do
        insrt := 0 ;
        for ex from 1 to nops(pi) do
            if AScompare(p, op(ex,pi)) > 0 then
                insrt := ex ;
            end if;
        end do:
        if nops(pi) = 0 then
            pi := [p] ;
        elif insrt = 0 then
            pi := [p,op(pi)] ;
        elif insrt = nops(pi) then
            pi := [op(pi),p] ;
        else
            pi := [op(1..insrt,pi),p,op(insrt+1..nops(pi),pi)] ;
        end if;
    end do:
    return pi ;
end proc:
A119441 := proc(n,k)
    local pi,a,p ;
    pi := ASPrts(n)[k] ;
    a := 1 ;
    for p in pi do
        a := a*combinat[numbpart](p) ;
    end do:
    a ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d,",A119441(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 12 2013

Formula

T(n,k) = product_{p=1..A036043(n,k)} A000041(c), 1<=k<=A000041(n), where c are the parts in the k-th partition of n. - R. J. Mathar, Jul 12 2013

A301595 Number of thrice-partitions of n.

Original entry on oeis.org

1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2018

Keywords

Comments

A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.

Examples

			The a(3) = 10 thrice-partitions:
  ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
  ((2))((1)), ((11))((1)), ((1)(1))((1)),
  ((1))((1))((1)).

Links

Crossrefs

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.
Column k=3 of A323718.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 25 2019
  • Mathematica
    twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
Array[thrie,30]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
     1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
a[n_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

Formula

O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019
Showing 1-4 of 4 results.

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