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 1-9 of 9 results.

A264609 A260685(4n).

Original entry on oeis.org

-2, -6, 4, -54, 28, 132, 124, -4470, 444, 5964, 2044, 89028, 8188, 248172, 9784, -30229110, 131068, 7314348, 524284, 361466316, 174136, 354424812, 8388604, 54096506052, 16344060, 12926541804, 46427196, 1198466847468, 536870908, 216322653432
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2015

Keywords

Crossrefs

Cf. A260685.

A279787 Twice-partitioned numbers where the first partition is constant.

Original entry on oeis.org

1, 1, 3, 4, 10, 8, 29, 16, 64, 58, 124, 57, 469, 102, 489, 763, 1597, 298, 3858, 491, 8942, 6355, 6187, 1256, 59076, 18766, 20830, 49694, 167078, 4566, 481186, 6843, 752128, 362907, 231592, 1597802, 5951007, 21638, 790404, 2655810, 25274798, 44584, 40898731
Offset: 0

Views

Author

Gus Wiseman, Dec 18 2016

Keywords

Examples

			The a(4)=10 twice-partitions are:
((4)), ((31)), ((22)), ((211)), ((1111)),
((2)(2)), ((2)(11)), ((11)(2)), ((11)(11)),
((1)(1)(1)(1)).

Links

Crossrefs

Cf. A000005, A000041, A018818, A063834, A260685, A271619.

Programs

  • Maple
    with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1,
      add(numbpart(n/d)^d, d=divisors(n)))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016
  • Mathematica
    nn=20;Table[DivisorSum[n,Power[PartitionsP[#],n/#]&],{n,nn}]
  • PARI
    a(n)=if(n==0, 1, sumdiv(n, d, numbpart(n/d)^d)) \\ Andrew Howroyd, Aug 26 2018

Formula

a(n) = Sum_{d|n} A000041(n/d)^d for n > 0. - Andrew Howroyd, Aug 26 2018

A281145 Number of same-trees of weight n.

Original entry on oeis.org

1, 2, 2, 6, 2, 14, 2, 54, 10, 38, 2, 494, 2, 134, 42, 4470, 2, 3422, 2, 10262, 138, 2054, 2, 490926, 34, 8198, 1514, 314294, 2, 628318, 2, 30229110, 2058, 131078, 162, 150147342, 2, 524294, 8202, 628073814, 2, 109952254, 2, 371210294, 207370, 8388614, 2
Offset: 1

Views

Author

Gus Wiseman, Jan 15 2017

Keywords

Comments

A same-tree is either: (case 1) a positive integer, or (case 2) a finite sequence of two or more same-trees all having the same weight, where the weight in case 2 is the sum of weights.

Examples

			The a(6)=14 same-trees are:
6,
(33),
(222),
(3(111)), ((111)3),
(22(11)), (2(11)2), ((11)22),
(2(11)(11)), ((11)2(11)), ((11)(11)2),
((111)(111)), ((11)(11)(11)), (111111).
The a(9)=10 same-trees are:
9,
(333),
(33(111)), (3(111)3), ((111)33),
(3(111)(111)), ((111)3(111)), ((111)(111)3),
((111)(111)(111)), (111111111).

Links

Crossrefs

Cf. A196545, A260685, A273873, A275870, A006241.

Programs

  • Mathematica
    a[n_]:=1+DivisorSum[n,b[#]^(n/#)&]-b[n]/.b->a;
Array[a,47]
  • PARI
    seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sumdiv(n, d, v[n/d]^d)); v} \\ Andrew Howroyd, Aug 20 2018

Formula

a(n) = 1 + Sum a(d)^(n/d) where the sum is over divisors less than n.

A279789 Number of ways to choose a constant partition of each part of a constant partition of n.

Original entry on oeis.org

1, 1, 3, 3, 8, 3, 17, 3, 30, 12, 41, 3, 130, 3, 137, 45, 359, 3, 656, 3, 1306, 141, 2057, 3, 5446, 36, 8201, 544, 18610, 3, 34969, 3, 72385, 2061, 131081, 165, 290362, 3, 524297, 8205, 1109206, 3, 2130073, 3, 4371490, 33594, 8388617, 3, 17445321, 132, 33556496
Offset: 0

Views

Author

Gus Wiseman, Dec 18 2016

Keywords

Comments

Also number of ways to choose a divisor d|n and then a sequence of n/d divisors of d.

Examples

			The a(6)=17 twice-constant partitions are:
((6)),
((3)(3)), ((33)),
((3)(111)), ((111)(3)),
((2)(2)(2)), ((222)),
((2)(2)(11)), ((2)(11)(2)), ((11)(2)(2)),
((2)(11)(11)), ((11)(2)(11)), ((11)(11)(2)),
((1)(1)(1)(1)(1)(1)), ((11)(11)(11)), ((111)(111)), ((111111)).

Links

Crossrefs

Cf: A000005 (tau), A000040, A018818, A063834, A260685, A279375.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(tau(n/d)^d, d=divisors(n)))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 20 2016
  • Mathematica
    nn=20;Table[DivisorSum[n,Power[DivisorSigma[0,#],n/#]&],{n,nn}]
  • PARI
    a(n)=if(n==0, 1, sumdiv(n, d, numdiv(n/d)^d)) \\ Andrew Howroyd, Aug 26 2018

Formula

a(n) = Sum_{d|n} tau(n/d)^d for n > 0. - Andrew Howroyd, Aug 26 2018
G.f.: 1 + Sum_{k>=1} tau(k)*x^k/(1 - tau(k)*x^k). - Ilya Gutkovskiy, May 23 2019
a(n) = 3 <=> n is prime <=> n in { A000040 }. - Alois P. Heinz, May 23 2019

A290971 Write x/(1-x) in the form Sum_{j>=1} a(j)*x^j/(1+a(j)*x^j).

Original entry on oeis.org

1, 2, 0, 6, 0, -6, 0, 54, 0, -30, 0, -114, 0, -126, 0, 4470, 0, -294, 0, -5850, 0, -2046, 0, -92418, 0, -8190, 0, -247674, 0, 2010, 0, 30229110, 0, -131070, 0, -8200914, 0, -524286, 0, -362617770, 0, 183162, 0, -354416634, 0, -8388606, 0, -53614489794, 0
Offset: 1

Views

Author

Gus Wiseman, Aug 16 2017

Keywords

Crossrefs

Cf. A048272, A220418, A260685, A273873, A275870, A281145, A289078, A289501, A290261, A290973.

Programs

  • Mathematica
    nn=20;-Solve[Table[Sum[a[n/d]^d,{d,Divisors[n]}]==-1,{n,nn}],Array[a,nn]][[1,All,2]]

Formula

a(n) = -Sum_t (-1)^v(t) where the sum is over all same-trees of weight n (see A281145 for definition) and v(t) is the number of nodes (branchings and leaves) in t.

A290973 Write 2*x/(1-x) in the form Sum_{j>=1} ((1-x^j)^a(j) - 1).

Original entry on oeis.org

-2, 1, 2, 3, 4, 6, 6, 10, 8, 15, 10, 25, 12, 28, 10, 60, 16, 25, 18, 125, 0, 66, 22, 218, 24, 91, -30, 420, 28, -387, 30, 2011, -88, 153, 28, -1894, 36, 190, -182, 8902, 40, -3234, 42, 2398, -132, 276, 46, 2340, 48, -2678, -510, 4641, 52, -1754, -198, 108400
Offset: 1

Views

Author

Gus Wiseman, Aug 16 2017

Keywords

Examples

			2x/(1-x) = (1-x)^(-2) - 1 + (1-x^2)^1 - 1 + (1-x^3)^2 - 1 + (1-x^4)^3 - 1 + ...

Crossrefs

Cf. A048272, A220418, A260685, A281145, A289078, A289501, A290261, A290971.

Programs

  • Maple
    a:= n-> add(binomial(n/d-1-a(d), n/d), d=
        numtheory[divisors](n) minus {n})-2:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 27 2017
  • Mathematica
    nn=60;
rus=SolveAlways[Normal[Series[2x/(1-x)==Sum[(1-x^n)^a[n]-1,{n,nn}],{x,0,nn}]],x];
Array[a,nn]/.First[rus]

Formula

For all n > 0 we have: 2 = Sum_{d|n} binomial(-a(d) + n/d - 1, n/d).

A382524 Number of ways to choose a different constant partition of each part of a constant partition of n.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 6, 2, 10, 3, 6, 2, 24, 2, 6, 4, 17, 2, 36, 2, 18, 4, 6, 2, 86, 3, 6, 10, 18, 2, 44, 2, 50, 4, 6, 4, 159, 2, 6, 4, 62, 2, 44, 2, 18, 30, 6, 2, 486, 3, 12, 4, 18, 2, 140, 4, 62, 4, 6, 2, 932, 2, 6, 30, 157, 4, 44, 2, 18, 4, 20, 2, 1500, 2, 6
Offset: 0

Views

Author

Gus Wiseman, Apr 03 2025

Keywords

Comments

These are strict twice-partitions of weight n and type PRR.

Examples

			The a(1) = 1 through a(8) = 10 twice-partitions:
  (1)  (2)   (3)    (4)      (5)      (6)       (7)        (8)
       (11)  (111)  (22)     (11111)  (33)      (1111111)  (44)
                    (1111)            (222)                (2222)
                    (11)(2)           (111111)             (22)(4)
                    (2)(11)           (111)(3)             (4)(22)
                                      (3)(111)             (1111)(4)
                                                           (4)(1111)
                                                           (11111111)
                                                           (1111)(22)
                                                           (22)(1111)

Links

Crossrefs

For distinct instead of equal block-sums we have A279786.
This is the strict case of A279789.
The orderless version is A304442, see A353833, A381995, A381871.
Multiset partitions of this type are ranked by A326534 /\ A355743 /\ A005117.
Partitions with no partition of this type are counted by A382076, strict case of A381993.
Normal multiset partitions of this type are counted by the strict case of A382204.
A006171 counts multiset partitions into constant blocks of integer partitions of n.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000005, A000040, A000688, A018818, A047966, A063834, A260685, A279784, A356065, A381453, A381455.

Programs

  • Mathematica
    Table[If[n==0,1,Sum[Binomial[Length[Divisors[n/d]],d]*d!,{d,Divisors[n]}]],{n,0,100}]

Formula

a(n) = Sum_{d|n} binomial(A000005(n/d),d) * d!

A264610 a(0)=1; thereafter a(n) = -Sum_{i=1..n} a(n-i)^(2^i).

Original entry on oeis.org

1, -1, -2, -6, -54, -4470, -30229110, -1388162822415990, -2926650008246700930753649260150, -13001299396349738096788020339357180951040927696209747704510070
Offset: 0

Views

Author

N. J. A. Sloane, Nov 25 2015

Keywords

Crossrefs

Equals A260685(2^n).

Programs

  • Maple
    f:=proc(n) option remember;
if n=0 then 1 else -add(f(n-i)^(2^i),i=1..n); fi; end;
[seq(f(n),n=0..4)];

A301761 Number of ways to choose a rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 13, 12, 26, 31, 57, 43, 150, 78, 224, 293, 484, 232, 1190, 386, 2260, 2087, 2558, 1003, 11154, 4701, 7889, 13597, 30041, 3719, 83248, 5605, 95006, 84486, 63506, 251487, 654394, 17978, 169864, 490741, 2290336, 37339, 4079503, 53175, 3979370
Offset: 1

Views

Author

Gus Wiseman, Mar 26 2018

Keywords

Comments

A rooted partition of n is an integer partition of n - 1.

Examples

			The a(7) = 13 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(2)(2), (2)(11), (11)(2), (11)(11),
(1)(1)(1),
()()()()()().

Links

Crossrefs

Cf. A000005, A000041, A002865, A018818, A063834, A093637, A127524, A260685, A271619, A279787, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    Table[Sum[PartitionsP[n/d-1]^d,{d,Divisors[n]}],{n,50}]
  • PARI
    a(n)=if(n==1, 1, sumdiv(n-1, d, numbpart((n-1)/d-1)^d)) \\ Andrew Howroyd, Aug 26 2018

Formula

a(n) = Sum_{d | n-1} A000041((n-1)/d-1)^d for n > 1. - Andrew Howroyd, Aug 26 2018
Showing 1-9 of 9 results.

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

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