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.

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

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

A036895 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

0, 0, 1, 4, 12, 29, 67, 146, 313, 654, 1342, 2691, 5291, 10184, 19237, 35680, 65102, 116967, 207241, 362423, 626265, 1070100, 1809513, 3029902, 5026778, 8267109, 13484325, 21821771, 35051459, 55901678, 88550088, 139355911, 217949589, 338837468
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Formula

a(n) = A036891(n) - A036892(n)
a(n) = A036887(n) - A202086(n)

Extensions

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

A036887 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, 10, 25, 62, 145, 323, 689, 1417, 2831, 5517, 10532, 19734, 36377, 66042, 118240, 208929, 364689, 629238, 1073964, 1814246, 3035236, 5031509, 8268583, 13476606, 21793642, 34981783, 55753411, 88258773, 138813831, 216978085, 337147547
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Formula

a(n) = A036882(n) - A036889(n)
a(n) = A202086(n) + A036895(n)

Extensions

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

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)

A202192 Number of partitions of 5n with equal number of parts congruent to each of 1, 2, 3 and 4 modulo 5.

Original entry on oeis.org

1, 1, 3, 8, 22, 53, 124, 269, 568, 1152, 2284, 4410, 8363, 15542, 28438, 51201, 90930, 159300, 275740, 471706, 798388, 1337478, 2219395, 3649432, 5950078, 9622364, 15442269, 24600952, 38919910, 61164114, 95513618, 148247892, 228761668, 351032568, 535772894
Offset: 0

Views

Author

Max Alekseyev, Dec 14 2011

Keywords

Links

Crossrefs

Cf. A046776.

Programs

  • 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}] + PartitionsP[n];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 25 2019, after Alois P. Heinz in A046787 *)

Formula

a(n) = A046776(n) + A202086(n) + A202088(n).
a(n) = A046787(n) + A000041(n).

Extensions

a(33)-a(34) from Alois P. Heinz, May 24 2019

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 1-7 of 7 results.

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