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.

A265250 Number of partitions of n having no parts strictly between the smallest and the largest part (n>=1).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 17, 20, 26, 29, 35, 39, 48, 48, 60, 61, 74, 73, 87, 86, 106, 99, 120, 112, 140, 130, 155, 143, 176, 159, 194, 180, 216, 186, 240, 209, 258, 234, 274, 243, 308, 261, 325, 289, 348, 297, 383, 314, 392, 356, 423, 355, 460, 372, 468, 422
Offset: 1

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Author

Emeric Deutsch, Dec 25 2015

Keywords

Examples

			a(3) = 3 because we have [3], [1,2], [1,1,1] (all partitions of 3).
a(6) = 10 because we have all A000041(6) = 11 partitions of 6 except [1,2,3].
a(7) = 13 because we have all A000041(7) = 15 partitions of 7 except [1,2,4] and [1,1,2,3].
		

Crossrefs

Programs

  • Maple
    g := add(x^i/(1-x^i), i = 1 .. 80)+add(add(x^(i+j)/((1-x^i)*(1-x^j)), j = i+1..80),i=1..80): gser := series(g,x=0,60): seq(coeff(gser,x,n),n=1..50);
    # second Maple program:
    b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0,
          `if`(t=1, `if`(irem(n, i)=0, 1, 0)+b(n, i-1, t),
           add(b(n-i*j, i-1, t-`if`(j=0, 0, 1)), j=0..n/i))))
        end:
    a:= n-> b(n$2, 2):
    seq(a(n), n=1..100);  # Alois P. Heinz, Jan 01 2016
  • Mathematica
    b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0, If[t == 1, If[Mod[n, i] == 0, 1, 0] + b[n, i - 1, t], Sum[b[n - i*j, i - 1, t - If[j == 0, 0, 1]], {j, 0, n/i}]]]]; a[n_] := b[n, n, 2]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

a(n) = A265249(n,0).
G.f.: G(x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} Sum_{j>=i+1} x^(i+j)/ ((1-x^i)*(1-x^j)).
a(n) = A116608(n,1) + A116608(n,2) = A000005(n) + A002133(n). - Seiichi Manyama, Sep 14 2023

A309097 Number of partitions of n avoiding the partition (4,2,1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 32, 39, 49, 56, 68, 79, 91, 103, 119, 132, 150, 165, 183, 202, 224, 241, 264, 287, 311, 334, 362, 385, 415, 442, 472, 503, 535, 563, 599, 634, 670, 703, 743, 778, 820, 859, 899, 942, 988, 1027, 1074, 1119, 1167, 1214, 1266
Offset: 0

Views

Author

Jonathan S. Bloom, Jul 12 2019

Keywords

Comments

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.
Conjecture: for n > 0, a(n) is the number of ordered pairs (r, l) such that there exists a nilpotent matrix of order n whose rank is r and nilpotent index is l. Actually, such a matrix exists if and only if ceiling(n/(n-r)) <= l <= r+1, see my proof below. If this conjecture is true, then a(n) = (n^2 + 3n)/2 - A006590(n) for n > 0. - Jianing Song, Nov 04 2019

Crossrefs

Programs

  • PARI
    lista(n)=my(b(k)=x^k/(1-x^k)+O(x*x^n));Vec(1+sum(i=1,n,b(i)*(1+sum(j=i+1,n,b(j)*(1+b(j+1)))))) \\ Christian Sievers, Sep 01 2025

Formula

G.f.: 1 + Sum_{i>=1} b(i) * ( 1 + Sum_{j>i} b(j) * ( 1 + b(j+1) ) ) where b(k)=x^k/(1-x^k). - Christian Sievers, Sep 01 2025

Extensions

More terms from Alois P. Heinz, Jul 12 2019

A309099 Number of partitions of n avoiding the partition (4,3,1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 37, 46, 59, 72, 87, 104, 124, 144, 168, 192, 220, 250, 282, 314, 352, 391, 432, 475, 522, 569, 622, 675, 732, 791, 852, 915, 985, 1055, 1127, 1201, 1281, 1361, 1447, 1533, 1623, 1717, 1813, 1909, 2013, 2118, 2227, 2338, 2453
Offset: 0

Views

Author

Jonathan S. Bloom, Jul 12 2019

Keywords

Comments

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Crossrefs

Programs

  • Maple
    b:= proc(n) option remember; `if`(n<1, [0$2],
          (p-> p+[numtheory[tau](n), p[1]])(b(n-1)))
        end:
    a:= n-> b(n+1)[2]+`if`(n=0, 1, n*(1-n)):
    seq(a(n), n=0..55);  # Alois P. Heinz, Dec 20 2023
  • Mathematica
    b[n_] := b[n] = If[n < 1, {0, 0}, With[{p = b[n-1]},
       p + {DivisorSigma[0, n], p[[1]]}]];
    a[n_] := b[n+1][[2]] + If[n == 0, 1, n*(1-n)];
    Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jan 29 2025, after Alois P. Heinz *)
  • PARI
    a(n) = if(n == 0, 1, sum(i = 1, n, (n - i + 1) * numdiv(i)) - n * (n - 1)) \\ Mikhail Kurkov, Dec 20 2023 [verification needed]

Formula

a(n) = A078567(n+1) - A002378(n-1) for n > 0 with a(0) = 1. - Mikhail Kurkov, Dec 20 2023 [verification needed]

Extensions

More terms from Alois P. Heinz, Jul 12 2019

A309098 Number of partitions of n avoiding the partition (4,3).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 33, 39, 51, 58, 72, 82, 99, 110, 131, 143, 168, 183, 210, 226, 259, 277, 312, 333, 372, 394, 439, 462, 511, 537, 588, 617, 675, 705, 765, 798, 864, 898, 970, 1005, 1081, 1121, 1199, 1240, 1326, 1369, 1459, 1505, 1599, 1646
Offset: 0

Views

Author

Jonathan S. Bloom, Jul 12 2019

Keywords

Comments

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Crossrefs

Programs

  • PARI
    lista(n)=Vec((1+sum(i=3,n,x^i/(1-x^i)+O(x*x^n)))/(1-x-x^2+x^3)) \\ Christian Sievers, Sep 01 2025

Formula

G.f.: (1 + Sum_{i>=3} x^i/(1-x^i)) / (1-x-x^2+x^3). - Christian Sievers, Sep 01 2025

Extensions

More terms from Alois P. Heinz, Jul 12 2019

A364809 Number of partitions of n with at most five part sizes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 791, 1000, 1250, 1565, 1938, 2400, 2945, 3615, 4395, 5342, 6439, 7755, 9268, 11069, 13127, 15537, 18286, 21484, 25095, 29275, 33968, 39344, 45362, 52193, 59836, 68441, 78014, 88724, 100622, 113828
Offset: 0

Views

Author

Seiichi Manyama, Sep 14 2023

Keywords

Crossrefs

Programs

  • Python
    from sympy.utilities.iterables import partitions
    def A364809(n): return sum(1 for p in partitions(n) if len(p)<=5) # Chai Wah Wu, Sep 14 2023

Formula

a(n) = Sum_{k=1..5} A116608(n,k).

A364793 Number of partitions of n with at most four part sizes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 175, 229, 292, 375, 470, 591, 733, 905, 1103, 1343, 1615, 1938, 2309, 2726, 3211, 3758, 4379, 5069, 5865, 6716, 7694, 8769, 9967, 11254, 12732, 14264, 16025, 17877, 19959, 22149, 24605, 27147, 30012, 33006, 36294, 39742, 43573, 47524
Offset: 0

Views

Author

Seiichi Manyama, Sep 14 2023

Keywords

Crossrefs

Programs

  • Python
    from sympy.utilities.iterables import partitions
    def A364793(n): return sum(1 for p in partitions(n) if len(p)<=4) # Chai Wah Wu, Sep 14 2023

Formula

a(n) = Sum_{k=1..4} A116608(n,k).

A309194 Number of partitions of n avoiding the partition (4,3,2).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 51, 67, 82, 105, 125, 154, 180, 218, 250, 295, 334, 390, 436, 502, 553, 630, 694, 780, 849, 950, 1027, 1138, 1230, 1355, 1447, 1590, 1694, 1846, 1971, 2133, 2257, 2445, 2579, 2776, 2932, 3142, 3298, 3539, 3702, 3941, 4139
Offset: 0

Views

Author

Jonathan S. Bloom, Jul 16 2019

Keywords

Comments

We say a partition alpha contains mu provided that one can delete rows and columns from (the Ferrers board of) alpha and then top/right justify to obtain mu. If this is not possible then we say alpha avoids mu. For example, the only partitions avoiding (2,1) are those whose Ferrers boards are rectangles.

Crossrefs

Extensions

More terms from Alois P. Heinz, Jul 18 2019
Showing 1-7 of 7 results.