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 1-7 of 7 results.

A015723 Number of parts in all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 3, 3, 5, 8, 10, 13, 18, 25, 30, 40, 49, 63, 80, 98, 119, 149, 179, 218, 266, 318, 380, 455, 541, 640, 760, 895, 1050, 1234, 1442, 1679, 1960, 2272, 2635, 3052, 3520, 4054, 4669, 5359, 6142, 7035, 8037, 9170, 10460, 11896, 13517, 15349, 17394, 19691
Offset: 1

Views

Author

Clark Kimberling

Keywords

Examples

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with a total of 1 + 2 + 2 + 3 = 8 parts, so a(6) = 8. - _Gus Wiseman_, May 09 2019

Links

Crossrefs

Cf. A006128, A008289, A079499, A067619, A186545.
Column k=1 of: A210485, A213177, A327622.
Row lengths of A325537.
Cf. A022629, A066186, A066189, A325504, A325505, A325506, A325513, A325515.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 27 2013
  • Mathematica
    nn=50; Rest[CoefficientList[Series[D[Product[1+y x^i,{i,1,nn}],y]/.y->1,{x,0,nn}],x]]  (* Geoffrey Critzer, Oct 29 2012; fixed by Vaclav Kotesovec, Apr 16 2016 *)
q[n_, k_] := q[n, k] = If[nVaclav Kotesovec, Apr 16 2016 *)
Table[Length[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,50}] (* Gus Wiseman, May 09 2019 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0},
   Sum[{#[[1]], #[[2]] + #[[1]]*j}&@ b[n-i*j, i-1], {j, 0, Min[n/i, 1]}]]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)
  • PARI
    N=66;  q='q+O('q^N); gf=sum(n=0,N, n*q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
Vec(gf) /* Joerg Arndt, Oct 20 2012 */

Formula

G.f.: sum(k>=1, x^k/(1+x^k) ) * prod(m>=1, 1+x^m ). Convolution of A048272 and A000009. - Vladeta Jovovic, Nov 26 2002
G.f.: sum(k>=1, k*x^(k*(k+1)/2)/prod(i=1..k, 1-x^i ) ). - Vladeta Jovovic, Sep 21 2005
a(n) = A238131(n)+A238132(n) = sum_{k=1..n} A048272(k)*A000009(n-k). - Mircea Merca, Feb 26 2014
a(n) = Sum_{k>=1} k*A008289(n,k). - Vaclav Kotesovec, Apr 16 2016
G.f.: -(-1; x)inf * (log(1-x) + psi_x(1 - log(-1)/log(x)))/(2*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - _Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (2 * Pi * n^(1/4)). - Vaclav Kotesovec, May 19 2018
For n > 0, a(n) = A116676(n) + A116680(n). - Vaclav Kotesovec, May 26 2018

Extensions

Extended and corrected by Naohiro Nomoto, Feb 24 2002

A066189 Sum of all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 2, 6, 8, 15, 24, 35, 48, 72, 100, 132, 180, 234, 308, 405, 512, 646, 828, 1026, 1280, 1596, 1958, 2392, 2928, 3550, 4290, 5184, 6216, 7424, 8880, 10540, 12480, 14784, 17408, 20475, 24048, 28120, 32832, 38298, 44520, 51660, 59892, 69230, 79904
Offset: 0

Views

Author

Wouter Meeussen, Dec 15 2001

Keywords

Examples

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)} with sum 6+5+1+4+2+3+2+1 = 24. - _Gus Wiseman_, May 09 2019

Links

Crossrefs

Row sums of A026793, A118457, A246688, A325537.
Cf. A015723, A022629, A066186, A147655, A325504, A325505, A325506, A325513, A325515, A325537.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i>n, [0$2],
      b(n, i+1)+(p-> p+[0, i*p[1]])(b(n-i, i+1))))
    end:
a:= n-> b(n, 1)[2]:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 01 2014
  • Mathematica
    PartitionsQ[ Range[ 60 ] ]Range[ 60 ]
nmax=60; CoefficientList[Series[x*D[Product[1+x^k, {k, 1, nmax}], x], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 21 2016 *)

Formula

G.f.: sum(n>=1, n*q^(n-1)/(1+q^n) ) * prod(n>=1, 1+q^n ). - Joerg Arndt, Aug 03 2011
a(n) = n * A000009(n). - Vaclav Kotesovec, Sep 25 2016
G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^k). - Vaclav Kotesovec, Nov 21 2016
a(n) = A056239(A325506(n)). - Gus Wiseman, May 09 2019

A325504 Product of products of parts over all strict integer partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 120, 1440, 40320, 1209600, 1567641600, 2633637888000, 13905608048640000, 5046067048690483200000, 5289893008483207348224000000, 1266933607446134946465526579200000000, 99304891373531545064656621572980736000000000000
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Examples

			The strict partitions of 5 are {(5), (4,1), (3,2)} with product a(5) = 5*4*1*3*2 = 120.
The sequence of terms together with their prime indices begins:
              1: {}
              1: {}
              2: {1}
              6: {1,2}
             12: {1,1,2}
            120: {1,1,1,2,3}
           1440: {1,1,1,1,1,2,2,3}
          40320: {1,1,1,1,1,1,1,2,2,3,4}
        1209600: {1,1,1,1,1,1,1,1,2,2,2,3,3,4}
     1567641600: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4}
  2633637888000: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4}

Links

Crossrefs

Cf. A000009, A006128, A007870 (non-strict version), A015723, A022629 (sum of products of parts), A066186, A066189, A066633, A246867, A325505, A325506, A325512, A325513, A325515.

Programs

  • Maple
    a:= n-> mul(i, i=map(x-> x[], select(x->
        nops(x)=nops({x[]}), combinat[partition](n)))):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0, 1], ((f, g)->
     [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021
  • Mathematica
    Table[Times@@Join@@Select[IntegerPartitions[n],UnsameQ@@#&],{n,0,10}]

Formula

A001222(a(n)) = A325515(n).
a(n) = A003963(A325506(n)).

A325506 Product of Heinz numbers over all strict integer partitions of n.

Original entry on oeis.org

1, 2, 3, 30, 70, 2310, 180180, 21441420, 6401795400, 200984366583000, 41615822944675980000, 10515527757483671302380000, 4919824049783476260137727416400000, 5158181210492841550866520676965246284000000, 29776760895364738730693151196801613158042403043600000000
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

a(n) is the product of row n of A246867 (squarefree numbers arranged by sum of prime indices).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with Heinz numbers {13,22,21,30}, with product 13*22*21*30 = 180180, so a(6) = 180180.
The sequence of terms together with their prime indices begins:
                     1: {}
                     2: {1}
                     3: {2}
                    30: {1,2,3}
                    70: {1,3,4}
                  2310: {1,2,3,4,5}
                180180: {1,1,2,2,3,4,5,6}
              21441420: {1,1,2,2,3,4,4,5,6,7}
            6401795400: {1,1,1,2,2,3,3,4,5,5,6,7,8}
       200984366583000: {1,1,1,2,2,2,3,3,3,4,4,5,5,6,6,7,8,9}
  41615822944675980000: {1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,6,6,7,7,8,9,10}

Crossrefs

Cf. A003963, A006128, A015723, A022629, A056239, A112798, A147655, A215366, A246867, A325501, A325504, A325505, A325512, A325513.

Programs

  • Mathematica
    Table[Times@@Prime/@(Join@@Select[IntegerPartitions[n],UnsameQ@@#&]),{n,0,15}]

Formula

a(n) = Product_{i = 1..A000009(n)} A246867(n,i).
A001222(a(n)) = A015723(n).
A056239(a(n)) = A066189(n).
A003963(a(n)) = A325504(n).
a(n) = A003963(A325505(n)).

A325513 Heinz number of the integer partition whose parts are the multiplicities in the multiset union of all strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 8, 8, 32, 144, 432, 2160, 27000, 582120, 7623000, 336936600, 6740402760, 543454231320, 57619849046760, 4683793138766280, 412882704970215480, 88171665744392750520, 12780536107937124847320, 2685589660883755945879560, 942036670625665177379096280
Offset: 0

Views

Author

Gus Wiseman, May 07 2019

Keywords

Comments

Also the Heinz number of row n of A015716 (with zeros removed).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), so a(6) = prime(1)^4*prime(2)^2 = 144.
The sequence of terms together with their prime indices begins:
               1: {}
               2: {1}
               2: {1}
               8: {1,1,1}
               8: {1,1,1}
              32: {1,1,1,1,1}
             144: {1,1,1,1,2,2}
             432: {1,1,1,1,2,2,2}
            2160: {1,1,1,1,2,2,2,3}
           27000: {1,1,1,2,2,2,3,3,3}
          582120: {1,1,1,2,2,2,3,4,4,5}
         7623000: {1,1,1,2,2,3,3,3,4,5,5}
       336936600: {1,1,1,2,2,3,3,4,5,5,6,7}
      6740402760: {1,1,1,2,2,3,4,4,4,6,6,7,8}
    543454231320: {1,1,1,2,2,3,4,4,5,6,7,8,9,10}
  57619849046760: {1,1,1,2,2,3,4,5,5,6,8,9,10,11,12}

Links

Crossrefs

Cf. A000009, A006128, A015716, A015723, A022629, A056239, A066633, A112798, A246867, A325500, A325504, A325506.

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n>(i*(i+1)/2), 0, `if`(n=0, [1, 0], b(n, i-1)+
          (p-> p+[0, p[1]*x^i])(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> (p-> mul((c-> `if`(c=0, 1, ithprime(c)))(
    coeff(p, x, i)), i=1..degree(p)))(b(n$2)[2]):
seq(a(n), n=0..21);  # Alois P. Heinz, Feb 23 2024
  • Mathematica
    Table[Times@@Prime/@Length/@Split[Sort[Join@@Select[IntegerPartitions[n],UnsameQ@@#&]]],{n,0,15}]

Formula

a(n) = A181819(A003963(A325505(n))).
A056239(a(n)) = A015723(n).

A325500 Heinz number of the set of Heinz numbers of integer partitions of n. Heinz numbers of rows of A215366.

Original entry on oeis.org

2, 3, 35, 2717, 22235779, 3163570326979, 51747966790650260753033, 188828800892079861898153036258130093, 2034903808706825942766196978067005215014684343665351270467, 75367279796373180679613801327275978589820813788234346991420766634058571423774287454563
Offset: 0

Views

Author

Gus Wiseman, May 05 2019

Keywords

Comments

The Heinz number of a set of positive integers {y_1,...,y_k} is prime(y_1)*...*prime(y_k).
All terms are squarefree and pairwise relatively prime.

Examples

			The integer partitions of 3 are {(3), (2,1), (1,1,1)}, with Heinz numbers {5,6,8}, with Heinz number prime(5)*prime(6)*prime(8) = 2717, so a(3) = 2717.
The sequence of terms together with their prime indices begins:
                        2: {1}
                        3: {2}
                       35: {3,4}
                     2717: {5,6,8}
                 22235779: {7,9,10,12,16}
            3163570326979: {11,14,15,18,20,24,32}
  51747966790650260753033: {13,21,22,25,27,28,30,36,40,48,64}

Links

Crossrefs

A subsequence of A005117.
Cf. A001222, A002110, A003963, A006128, A007870, A056239, A066186, A066633, A112798, A145519, A215366, A325501, A325505, A325507.

Programs

  • Mathematica
    Table[Times@@Prime/@(Times@@Prime/@#&/@IntegerPartitions[n]),{n,0,5}]

Formula

A001221(a(n)) = A001222(a(n)) = A000041(n).
A056239(a(n)) = A145519(n).
A003963(a(n)) = A325501(n).
A181819(A003963(a(n))) = A325507(n).

A325502 Heinz number of row n of Pascal's triangle A007318.

Original entry on oeis.org

2, 4, 12, 100, 2548, 407044, 106023164, 136765353124, 399090759725236, 4445098474836287524, 151287513513627682258436, 12698799587219706700017036196, 3463928752077516667634331415766516, 2591202267595530693505786197581910681796
Offset: 0

Views

Author

Gus Wiseman, May 06 2019

Keywords

Comments

The Heinz number of a positive integer sequence (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Every odd-indexed term is a square of a squarefree number.

Examples

			Row n = 5 of Pascal's triangle is (1,5,10,10,5,1), with Heinz number prime(1)*prime(5)*prime(10)*prime(10)*prime(5)*prime(1) = 407044, so a(5) = 407044.
The sequence of terms together with their prime indices begins:
                    2: {1}
                    4: {1,1}
                   12: {1,1,2}
                  100: {1,1,3,3}
                 2548: {1,1,4,4,6}
               407044: {1,1,5,5,10,10}
            106023164: {1,1,6,6,15,15,20}
         136765353124: {1,1,7,7,21,21,35,35}
      399090759725236: {1,1,8,8,28,28,56,56,70}
  4445098474836287524: {1,1,9,9,36,36,84,84,126,126}

Links

Crossrefs

Cf. A000040, A001222, A001405, A007318, A056239, A112798, A145519, A215366, A325500, A325503, A325505, A325514.

Programs

  • Mathematica
    Times@@@Table[Prime[Binomial[n,k]],{n,0,5},{k,0,n}]

Formula

A061395(a(n)) = A001405(n).
A056239(a(n)) = A000079(n).
A181819(a(n)) = A038754(n + 1).
Showing 1-7 of 7 results.

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