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.

A096401 Number of balanced partitions of n into distinct parts: least part is equal to number of parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 26, 28, 32, 35, 39, 43, 48, 53, 59, 65, 72, 80, 88, 97, 107, 118, 129, 142, 155, 171, 186, 204, 222, 244, 265, 290, 315, 345, 374, 409, 443, 484, 524, 571, 618, 673, 727, 790
Offset: 1

Views

Author

Vladeta Jovovic, Aug 06 2004

Keywords

Examples

			a(14)=3 because we have 12+2, 7+4+3 and 6+5+3.

Links

Crossrefs

Cf. A000009, A025157, A237976, A237977, A237979.
Cf. A006141, A047993, A339446.

Programs

  • Maple
    G:=sum((x^(m*(3*m-1)/2)-x^(m*(3*m+1)/2))/product(1-x^i,i=1..m),m=1..20): Gser:=series(G,x=0,80): seq(coeff(Gser,x^n),n=1..78); # Emeric Deutsch, Mar 29 2005
  • PARI
    my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, x^(k*(3*k-1)/2)/prod(j=1, k-1, 1-x^j))) \\ Seiichi Manyama, Jan 15 2022

Formula

G.f.: Sum_{m>=1} (x^(m*(3*m-1)/2)-x^(m*(3*m+1)/2))/Product_{i=1..m} (1-x^i).
a(n) = A025157(n) - A237979(n) = A237977(n) - A237976(n) for n > 0. - Seiichi Manyama, Jan 13 2022
a(n) ~ (1 - A263719) * A025157(n). - Vaclav Kotesovec, Jan 15 2022

Extensions

More terms from Emeric Deutsch, Mar 29 2005

A039899 Number of partitions satisfying 0 < cn(0,5) + cn(2,5) + cn(3,5).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 8, 12, 18, 25, 36, 49, 68, 91, 123, 162, 214, 278, 362, 464, 596, 757, 961, 1209, 1521, 1897, 2366, 2931, 3627, 4463, 5487, 6711, 8200, 9976, 12121, 14672, 17738, 21371, 25716, 30852, 36964, 44168, 52709, 62746, 74600, 88497
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: o < 0 + 2 + 3 (OMZBBp).
Number of partitions of n such that (greatest part) > (multiplicity of greatest part), for n >= 1. For example, a(6) counts these 8 partitions: 6, 51, 42, 411, 33, 321, 3111, 21111. See the Mathematica program at A240057 for the sequence as a count of partitions defined in this manner, and related sequences. - Clark Kimberling, Apr 02 2014

Links

Crossrefs

Cf. A003106, A003114, A039900, A237976.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i<1, 0, b(n, i-1, t)+ `if`(i>n, 0, b(n-i, i,
      `if`(irem(i, 5) in {1, 4}, t, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 03 2014
  • Mathematica
    Table[Count[IntegerPartitions[n], p_ /; Min[p] < Length[p]], {n, 24}] (* Clark Kimberling, Feb 13 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<1, 0, b[n, i-1, t] + If[i > n, 0, b[n-i, i, If[MemberQ[{1, 4}, Mod[i, 5]], t, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)
  • PARI
    my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=0, N, x^k*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

Formula

G.f.: Sum_{k>=0} x^k * (1-x^(k*(k-1))) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022

A237979 Number of strict partitions of n such that (least part) > number of parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 22, 25, 28, 32, 35, 40, 45, 50, 56, 63, 70, 78, 87, 96, 107, 118, 131, 144, 160, 175, 194, 213, 235, 257, 284, 310, 342, 373, 410, 447, 491, 534, 585, 637, 696, 756, 826, 896, 977, 1060, 1153, 1250, 1359, 1471, 1597, 1729, 1874, 2026, 2195, 2371, 2565
Offset: 1

Views

Author

Clark Kimberling, Feb 18 2014

Keywords

Comments

Also the number of partitions into distinct parts with minimal part >= 2 and difference between parts >= 3. [Joerg Arndt, Mar 31 2014]

Examples

			a(9) = 3 counts these partitions:  9, 63, 54.

Links

Crossrefs

Cf. A000009, A237976, A096401, A237977, A025157.

Programs

  • Mathematica
    z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)
  • PARI
    N=66; q='q+O('q^N); Vec(-1+sum(n=0, N, q^(n*(3*n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

Formula

G.f. with a(0)=0: sum(n>=0, q^(n*(3*n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Mar 09 2014]
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1 + 3*r^2)) * n^(3/4)), where r = A263719 and c = 3*(log(r))^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 15 2022
a(n) ~ A263719 * A025157(n). - Vaclav Kotesovec, Jan 15 2022

A237977 Number of strict partitions of n such that (least part) <= number of parts.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 29, 36, 42, 51, 60, 72, 84, 100, 117, 137, 160, 187, 216, 251, 290, 334, 385, 442, 507, 581, 664, 757, 864, 982, 1116, 1266, 1435, 1622, 1835, 2069, 2333, 2626, 2954, 3316, 3724, 4172, 4673, 5227, 5844
Offset: 0

Views

Author

Clark Kimberling, Feb 18 2014

Keywords

Examples

			a(8) = 4 counts these partitions:  71, 53, 521, 431.

Links

Crossrefs

Cf. A000009, A237976, A096401, A237979, A025157, A039900.

Programs

  • Mathematica
    z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)
  • PARI
    my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ Seiichi Manyama, Jan 13 2022

Formula

G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - Seiichi Manyama, Jan 13 2022
a(n) = A000009(n) - A237979(n). - Vaclav Kotesovec, Jan 18 2022

Extensions

Prepended a(0)=0, Seiichi Manyama, Jan 13 2022
Showing 1-4 of 4 results.

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