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 1-10 of 12 results. Next

A046765 Number of partitions of n with equal number of parts congruent to each of 0, 1 and 2 (mod 3).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 7, 0, 0, 13, 0, 0, 25, 0, 0, 43, 0, 0, 77, 0, 0, 130, 0, 0, 222, 0, 0, 365, 0, 0, 603, 0, 0, 966, 0, 0, 1546, 0, 0, 2425, 0, 0, 3783, 0, 0, 5813, 0, 0, 8884, 0, 0, 13411, 0, 0, 20130, 0, 0, 29922, 0, 0, 44217, 0, 0, 64814, 0, 0, 94485, 0, 0
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Other similar sequences include:
Mod 4: A046766, A046767, A046768, A046769, A046770.
Mod 5: A046771, A046772, A046773, A046774, A046775, A046776.

Programs

  • Mathematica
    Table[Length[Select[Last /@ Transpose /@ Tally /@ Mod[IntegerPartitions[n], 3], Length[#] == 3 && Length[Union[#]] == 1 &]], {n, 50}] (* Jayanta Basu, Jun 28 2013 *)
  • PARI
    seq(n)={Vec(sum(k=0, n\6, x^(6*k)/prod(j=1, k, 1 - x^(3*j) + O(x*x^n))^3) + O(x*x^n))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: Sum_{k>=0} x^(6*k)/(Product_{j=1..k} 1 - x^(3*j))^3. - Andrew Howroyd, Sep 16 2019

A046787 Number of partitions of 5n with equal nonzero number of parts congruent to each of 1, 2, 3 and 4 modulo 5.

Original entry on oeis.org

0, 0, 1, 5, 17, 46, 113, 254, 546, 1122, 2242, 4354, 8286, 15441, 28303, 51025, 90699, 159003, 275355, 471216, 797761, 1336686, 2218393, 3648177, 5948503, 9620406, 15439833, 24597942, 38916192, 61159549, 95508014, 148241050, 228753319, 351022425, 535760584
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

Number of partitions of m with equal numbers of parts congruent to each of 1, 2, 3 and 4 (mod 5) is 0 unless m == 0 mod 5.

Links

Crossrefs

Other similar sequences include:
Mod 4: A046778, A046779, A046780, A046781, A046782.
Mod 5: A046783, A046784, A046785, A046786.
Cf. A046765, A046776, A202192.

Programs

  • Maple
    mkl:= proc(i,l) local ll, mn, x; ll:= `if`(irem(i, 5)=0, l, applyop(x->x+1, irem(i,5), l)); mn:= min(l[])-1; `if`(mn<=0, ll, map(x->x-mn, ll)) end:
g:= proc(n,i,t) local m, mx; if n<0 then 0 elif n=0 then `if`(t[1]>0 and t[1]=t[2] and t[2]=t[3] and t[3]=t[4], 1, 0) elif i=0 then 0 elif i<5 then mx:= max(t[]); m:= n-10*mx +t[1] +t[2]*2 +t[3]*3 +t[4]*4; `if`(m>=0 and irem(m, 10)=0, 1, 0) else g(n,i,t):= g(n, i-1, t) + g(n-i, i, mkl(i, t)) fi end:
a:= n-> g(5*n, 5*n, [0,0,0,0]):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 04 2009
  • Mathematica
    mkl[i_, l_] := Module[{ll, mn, x}, ll = If[Mod[i, 5] == 0, l, MapAt[#+1&, l, Mod[i, 5]]]; mn = Min[l]-1; If[mn <= 0, ll, Map[#-mn&, ll]]];
g[n_, i_, t_] := g[n, i, t] = Module[{m, mx}, If[n<0, 0, If[n==0, If[ t[[1]]>0 && Equal @@ t[[1;;4]], 1, 0], If[i==0, 0, If[i<5, mx = Max[t]; m = n - 10 mx + t[[1]] + 2 t[[2]] + 3 t[[3]] + 4 t[[4]]; If[m >= 0 && Mod[m, 10]==0, 1, 0], g[n, i-1, t] + g[n-i, i, mkl[i, t]]]]]]];
a[n_] := g[5n, 5n, {0, 0, 0, 0}];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 25 2019, after Alois P. Heinz *)
  • PARI
    seq(n)={Vec(sum(k=1, n\2, x^(2*k)/prod(j=1, k, 1 - x^j + O(x*x^(n-2*k)))^4)/prod(j=1, n, 1 - x^j + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

a(n) = A046776(n) + A202086(n) + A202088(n) - A000041(n) = A202192(n) - A000041(n). - Max Alekseyev
G.f.: (Sum_{k>0} x^(2*k)/(Product_{j=1..k} 1 - x^j)^4)/(Product_{j>0} 1 - x^j). - Andrew Howroyd, Sep 16 2019

Extensions

a(17)-a(32) from Alois P. Heinz, Jul 04 2009
a(33)-a(34) from Alois P. Heinz, Aug 13 2013

A036889 Number of partitions of 5n such that cn(1,5) = cn(4,5) <= cn(0,5) = cn(2,5) = cn(3,5).

Original entry on oeis.org

0, 1, 4, 12, 29, 66, 137, 279, 546, 1057, 2000, 3746, 6886, 12508, 22360, 39477, 68736, 118309, 201207, 338672, 564211, 931342, 1523628, 2472228, 3979651, 6359094, 10088975, 15899507, 24894711, 38740189, 59929503, 92185390, 141029958, 214628608
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) <= cn(0,5) = cn(1,5) = cn(4,5).
For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Formula

a(n) = A036882(n) - A036887(n)
a(n) = A036881(n) - A036885(n)
a(n) = A046776(n) + A036892(n)

Extensions

Terms a(10) onward from Max Alekseyev, Dec 11 2011

A202088 Number of partitions of 5n such that cn(0,5) < cn(1,5) = cn(4,5) = cn(2,5) = cn(3,5).

Original entry on oeis.org

0, 0, 1, 4, 11, 25, 55, 116, 245, 505, 1026, 2030, 3936, 7450, 13837, 25210, 45206, 79831, 139136, 239471, 407582, 686346, 1144532, 1890837, 3096692, 5029412, 8104448, 12961576, 20582130, 32459992, 50859769, 79192204, 122572743
Offset: 0

Views

Author

Max Alekseyev, Dec 11 2011

Keywords

Comments

For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Crossrefs

Cf. A036888, A036893, A046776, A202087.

Programs

  • PARI
    seq(n)={Vec(sum(k=0, n\2, x^(2*k)*(1-x^k)/prod(j=1, k, 1 - x^j + O(x*x^n))^5) + O(x*x^n), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

a(n) = A036888(n) - A036893(n).
a(n) = A202087(n) - A046776(n).
G.f.: Sum_{k>=0} x^(2*k)*(1-x^k)/(Product_{j=1..k} 1 - x^j)^5. - Andrew Howroyd, Sep 16 2019

Extensions

a(0)=0 prepended by Andrew Howroyd, Sep 16 2019

A036884 Number of partitions of 5n such that cn(0,5) = cn(1,5) = cn(4,5) <= cn(2,5) = cn(3,5).

Original entry on oeis.org

1, 3, 7, 18, 42, 97, 207, 431, 861, 1685, 3216, 6042, 11139, 20248, 36245, 64041, 111663, 192432, 327803, 552593, 922129, 1524496, 2497868, 4058745, 6542497, 10467325, 16626651, 26231148, 41114412, 64042922, 99164091, 152671363, 233762167
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Alternatively, number of partitions of 5n such that cn(0,5) = cn(2,5) = cn(3,5) <= cn(1,5) = cn(4,5).
For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Formula

a(n) = A036880(n) - A036888(n)
a(n) = A046776(n) + A036886(n)
a(n) = A036881(n) - A036890(n)

Extensions

Terms a(10) onward from Max Alekseyev, Dec 10 2011

A036886 Number of partitions of 5n such that cn(0,5) = cn(1,5) = cn(4,5) < cn(2,5) = cn(3,5).

Original entry on oeis.org

1, 3, 6, 13, 27, 61, 132, 285, 590, 1190, 2325, 4441, 8288, 15197, 27394, 48679, 85332, 147790, 253016, 428602, 718696, 1193779, 1964996, 3206966, 5191350, 8339001, 13296592, 21053380, 33112242, 51746168, 80372146, 124104612, 190557592
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Alternatively, number of partitions of 5n such that cn(0,5) = cn(2,5) = cn(3,5) < cn(1,5) = cn(4,5).
For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Formula

a(n) = A036884(n) - A046776(n)
a(n) = A036885(n) - A036894(n)
a(n) = A036883(n) - A036893(n)

Extensions

Terms a(10) onward from Max Alekseyev, Dec 10 2011

A036892 Number of partitions of 5n such that cn(1,5) = cn(4,5) < cn(0,5) = cn(2,5) = cn(3,5).

Original entry on oeis.org

0, 1, 3, 7, 14, 30, 62, 133, 275, 562, 1109, 2145, 4035, 7457, 13509, 24115, 42405, 73667, 126420, 214681, 360778, 600625, 990756, 1620449, 2628504, 4230770, 6758916, 10721739, 16892541, 26443435, 41137558, 63618639, 97825383, 149605621, 227593695
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) < cn(0,5) = cn(1,5) = cn(4,5).
For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Formula

a(n) = A036891(n) - A036895(n)
a(n) = A036890(n) - A036894(n)
a(n) = A036889(n) - A046776(n)

Extensions

Terms a(10) onward from Max Alekseyev, Dec 11 2011

A202086 Number of partitions of 5n such that cn(1,5) = cn(4,5) = cn(2,5) = cn(3,5) < cn(0,5).

Original entry on oeis.org

1, 2, 3, 6, 13, 33, 78, 177, 376, 763, 1489, 2826, 5241, 9550, 17140, 30362, 53138, 91962, 157448, 266815, 447699, 744146, 1225723, 2001607, 3241805, 5209497, 8309317, 13160012, 20701952, 32357095, 50263743, 77622174, 119197958
Offset: 1

Views

Author

Max Alekseyev, Dec 11 2011

Keywords

Comments

For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Formula

a(n) = A202085(n) - A046776(n)
a(n) = A036887(n) - A036895(n)

A202087 Number of partitions of 5n such that cn(0,5) <= cn(1,5) = cn(4,5) = cn(2,5) = cn(3,5).

Original entry on oeis.org

1, 0, 1, 5, 16, 40, 91, 191, 391, 776, 1521, 2921, 5537, 10301, 18888, 34061, 60568, 106162, 183778, 314258, 531573, 889779, 1475249, 2423709, 3948471, 6380559, 10232772, 16291635, 25759898, 40462162, 63156523, 97984149, 151139494
Offset: 0

Views

Author

Max Alekseyev, Dec 11 2011

Keywords

Comments

For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Crossrefs

Cf. A036880, A036883, A046776, A202088.

Programs

  • PARI
    seq(n)={Vec(sum(k=0, n\2, x^(2*k)/prod(j=1, k, 1 - x^j + O(x*x^n))^5) + O(x*x^n), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

a(n) = A036880(n) - A036883(n).
a(n) = A046776(n) + A202088(n).
G.f.: Sum_{k>=0} x^(2*k)/(Product_{j=1..k} 1 - x^j)^5. - Andrew Howroyd, Sep 16 2019

Extensions

a(0)=1 prepended by Andrew Howroyd, Sep 16 2019

A202085 Number of partitions of 5n such that cn(1,5) = cn(4,5) = cn(2,5) = cn(3,5) <= cn(0,5).

Original entry on oeis.org

1, 2, 4, 11, 28, 69, 153, 323, 647, 1258, 2380, 4427, 8092, 14601, 25991, 45724, 79469, 136604, 232235, 390806, 651132, 1074863, 1758595, 2853386, 4592952, 7337821, 11639376, 18337780, 28704122, 44653849, 69055688, 106188925, 162402533
Offset: 1

Views

Author

Max Alekseyev, Dec 11 2011

Keywords

Comments

For a given partition, cn(i,n) means the number of its parts equal to i modulo n.

Links

Formula

a(n) = A046776(n) + A202086(n)
a(n) = A036882(n) - A036891(n)
Showing 1-10 of 12 results. Next

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