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 1-5 of 5 results.

A046776 Number of partitions of 5n with equal number of parts congruent to each of 0, 1, 2, 3 and 4 (mod 5).

Original entry on oeis.org

1, 0, 0, 1, 5, 15, 36, 75, 146, 271, 495, 891, 1601, 2851, 5051, 8851, 15362, 26331, 44642, 74787, 123991, 203433, 330717, 532872, 851779, 1351147, 2128324, 3330059, 5177768, 8002170, 12296754, 18791945, 28566751, 43204575, 65022987, 97395386, 145217908
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

Cf. A046765, A046787.

Programs

  • Maple
    mkl:= proc(i,l) local ll, mn, ii, x; ii:= irem(i,5); ii:= `if`(ii=0, 5, ii); ll:= applyop(x->x+1, ii, l); mn:= min(l[]); `if`(mn=0, ll, map (x->x-mn, ll)) end:
g:= proc(n,i,t) local m, mx, j; if n<0 then 0 elif n=0 then `if`(nops ({t[]})=1, 1, 0) elif i=0 then 0 elif i<6 then mx:= max (t[]); m:= n-15*mx +add(t[j]*j, j=1..5); g(n,i,t):= `if`(m>=0 and irem(m, 15)=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$5]):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 04 2009
  • Mathematica
    $RecursionLimit = 1000; mkl[i_, l_List] := Module[{ ll, mn, ii, x}, ii = Mod[i, 5]; ii = If[ii == 0, 5, ii]; ll = MapAt[#+1&, l, ii]; mn = Min[l]; If[mn == 0, ll, Map [#-mn&, ll]]]; g[n_, i_, t_List] := g[n, i, t] = Module[{ m, mx, j}, Which[n<0, 0 , n == 0, If[Length[t // Union] == 1, 1, 0], i==0, 0, i<6, mx = Max[t]; m = n-15*mx + Sum[t[[j]]*j, {j, 1, 5}]; If[m >= 0 && Mod[m, 15] == 0, 1, 0], True, g[n, i-1, t] + g[n-i, i, mkl[i, t]]]]; a[n_] := g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 21 2015, after Alois P. Heinz *)
  • PARI
    seq(n)={Vec(sum(k=0, n\3, x^(3*k)/prod(j=1, k, 1 - x^j + O(x*x^n))^5) + O(x*x^n))} \\ Andrew Howroyd, Sep 16 2019

Formula

a(n) = A202085(n) - A202086(n).
a(n) = A036884(n) - A036886(n).
a(n) = A036889(n) - A036892(n).
a(n) = A202087(n) - A202088(n).
G.f.: Sum_{k>=0} x^(3*k)/(Product_{j=1..k} 1 - x^j)^5. - Andrew Howroyd, Sep 16 2019

Extensions

a(18)-a(35) from Alois P. Heinz, Jul 04 2009
Edited by Max Alekseyev, Dec 11 2011
a(36) from Alois P. Heinz, Feb 03 2013

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

A036880 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, 4, 12, 34, 85, 203, 454, 985, 2060, 4205, 8363, 16298, 31103, 58319, 107471, 195037, 348795, 615550, 1072706, 1847867, 3148444, 5309948, 8869172, 14680261, 24090035, 39210436, 63327665, 101527253, 161626560, 255579456, 401556210, 627039569, 973374176
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

Programs

  • Maple
    mkl:= proc(i,l) local ll, mn, x; ll:= applyop(x->x+1, irem(i,5)+1, l); mn:= min(ll[]); `if`(mn=0, ll, map(x->x-mn, ll)) end:
g:= proc (n,i,t) if n<0 then 0 elif n=0 then `if`(t[1]<=t[2] and t[2]=t[5] and t[5]<=t[3] and t[3]=t[4], 1, 0) elif i=0 then 0 elif i=1 then g(0, 0, [t[1], t[2]+n, t[3], t[4], t[5]]) elif i=2 then `if`(t[3]>t[4], 0, g(n-2*(t[4]-t[3]), 1, [t[1], t[2], t[4], t[4], t[5]])) 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, [0,0,0,0,0]):
seq(a(n), n=1..15);  # Alois P. Heinz, Jul 02 2009
  • Mathematica
    mkl[i_, l_] := Module[{ll, mn, x}, ll = MapAt[#+1&, l, Mod[i, 5]+1]; mn = Min[ll]; If[mn==0, ll, Map[#-mn&, ll]]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0 , If[t[[1]] <= t[[2]] && t[[2]] == t[[5]] && t[[5]] <= t[[3]] && t[[3]] == t[[4]], 1, 0], i==0, 0, i==1, g[0, 0, {t[[1]], t[[2]]+n, t[[3]], t[[4]], t[[5]]}] , i==2, If[t[[3]]>t[[4]], 0, g[n-2*(t[[4]]-t[[3]]), 1, {t[[1]], t[[2]], t[[4]], t[[4]], t[[5]]}]], True, g[n, i-1, t] + g[n-i, i, mkl[i, t]]]; a[n_] := g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Formula

a(n) = A202087(n) + A036883(n)
a(n) = A036884(n) + A036888(n)

Extensions

a(10)-a(31) from Alois P. Heinz, Jul 02 2009
Edited by Max Alekseyev, Dec 11 2011
a(32)-a(33) from Alois P. Heinz, Mar 12 2016

A036883 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, 45, 112, 263, 594, 1284, 2684, 5442, 10761, 20802, 39431, 73410, 134469, 242633, 431772, 758448, 1316294, 2258665, 3834699, 6445463, 10731790, 17709476, 28977664, 47036030, 75767355, 121164398, 192422933, 303572061, 475900075
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) - A202087(n)
a(n) = A036886(n) + A036893(n)

Extensions

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

A202091 Number of partitions of 5n such that cn(1,5) = cn(4,5) and cn(2,5) = cn(3,5).

Original entry on oeis.org

1, 3, 11, 32, 88, 221, 532, 1213, 2672, 5676, 11724, 23568, 46315, 89076, 168124, 311763, 569000, 1023128, 1814776, 3178000, 5499588, 9411392, 15938221, 26726372, 44402336, 73121988, 119418609, 193488816, 311150404, 496783420, 787753316
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. A046776, A036880, A036881, A036882, A036883, A036884, A036885, A036886, A036887, A036888, A036889, A036890, A036891, A036892, A036893, A036894, A036895, A202085, A202086, A202087, A202088

Formula

a(n) = A046776(n) + A202086(n) + A202088(n) + 2*( A036886(n) + A036892(n) + A036893(n) + A036894(n) + A036895(n) )
a(n) = A202192(n) + 2*( A036886(n) + A036892(n) + A036893(n) + A036894(n) + A036895(n) )
Showing 1-5 of 5 results.

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