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 10 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

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

A046778 Number of partitions of n with equal nonzero number of parts congruent to each of 0, 1 and 2 (mod 4).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 1, 5, 6, 1, 5, 13, 11, 5, 16, 30, 20, 16, 41, 59, 37, 45, 92, 111, 73, 107, 189, 198, 148, 236, 363, 351, 300, 481, 667, 615, 597, 936, 1186, 1079, 1160, 1737, 2058, 1889, 2189, 3127, 3508, 3303, 4020, 5465, 5903, 5745, 7193
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\7, x^(7*k)/prod(j=1, k, 1 - x^(4*j) + O(x*x^(n-7*k)))^3)/prod(j=0, n\4, 1 - x^(4*j+3) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(7*k)/(Product_{j=1..k} 1 - x^(4*j))^3)/(Product_{j>=0} 1 - x^(4*j+3)). - Andrew Howroyd, Sep 16 2019

A046779 Number of partitions of n with equal nonzero number of parts congruent to each of 0, 1 and 3 (mod 4).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 5, 0, 12, 0, 16, 0, 30, 0, 41, 0, 70, 0, 95, 0, 150, 0, 203, 0, 309, 0, 413, 0, 608, 0, 807, 0, 1161, 0, 1529, 0, 2154, 0, 2819, 0, 3911, 0, 5086, 0, 6951, 0, 8994, 0, 12146, 0, 15633, 0, 20881, 0, 26751, 0, 35392, 0, 45137, 0, 59197
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\8, x^(8*k)/prod(j=1, k, 1 - x^(4*j) + O(x*x^(n-8*k)))^3)/prod(j=0, n\4, 1 - x^(4*j+2) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(8*k)/(Product_{j=1..k} 1 - x^(4*j))^3)/(Product_{j>=0} 1 - x^(4*j+2)). - Andrew Howroyd, Sep 16 2019

A046780 Number of partitions of n with equal nonzero number of parts congruent to each of 0, 2 and 3 (mod 4).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 4, 5, 5, 5, 11, 16, 17, 17, 27, 40, 45, 46, 61, 90, 106, 111, 133, 187, 227, 243, 276, 372, 459, 503, 555, 713, 887, 989, 1078, 1333, 1656, 1877, 2039, 2437, 3008, 3449, 3755, 4376, 5345, 6185, 6765, 7731, 9324, 10844
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\9, x^(9*k)/prod(j=1, k, 1 - x^(4*j) + O(x*x^(n-9*k)))^3)/prod(j=0, n\4, 1 - x^(4*j+1) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(9*k)/(Product_{j=1..k} 1 - x^(4*j))^3)/(Product_{j>=0} 1 - x^(4*j+1)). - Andrew Howroyd, Sep 16 2019

A046781 Number of partitions of n with equal nonzero number of parts congruent to each of 1, 2 and 3 (mod 4).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 11, 0, 4, 0, 26, 0, 14, 0, 55, 0, 38, 0, 110, 0, 94, 0, 212, 0, 209, 0, 397, 0, 441, 0, 729, 0, 878, 0, 1320, 0, 1685, 0, 2357, 0, 3121, 0, 4160, 0, 5633, 0, 7258, 0, 9923, 0, 12518, 0, 17153, 0, 21346, 0, 29133, 0, 35998, 0, 48766
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\6, x^(6*k)/prod(j=1, k, 1 - x^(4*j) + O(x*x^(n-6*k)))^3)/prod(j=0, n\4, 1 - x^(4*j+4) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(6*k)/(Product_{j=1..k} 1 - x^(4*j))^3)/(Product_{j>=0} 1 - x^(4*j+4)). - Andrew Howroyd, Sep 16 2019

A046783 Number of partitions of n with equal nonzero number of parts congruent to each of 0, 1, 2 and 3 (mod 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 1, 5, 10, 1, 1, 5, 15, 21, 5, 5, 16, 36, 41, 19, 17, 41, 76, 80, 53, 46, 92, 151, 156, 132, 111, 192, 287, 307, 293, 248, 378, 537, 599, 616, 521, 722, 990, 1158, 1220, 1051, 1346, 1818, 2191, 2339, 2050, 2481, 3302
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\11, x^(11*k)/prod(j=1, k, 1 - x^(5*j) + O(x*x^(n-11*k)))^4)/prod(j=0, n\5, 1 - x^(5*j+4) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(11*k)/(Product_{j=1..k} 1 - x^(5*j))^3)/(Product_{j>=0} 1 - x^(5*j+4)). - Andrew Howroyd, Sep 16 2019

A046784 Number of partitions of n with equal nonzero number of parts congruent to each of 0, 1, 2 and 4 (mod 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 4, 1, 0, 5, 1, 10, 5, 2, 15, 5, 22, 16, 9, 37, 16, 45, 42, 30, 80, 44, 91, 96, 80, 162, 105, 183, 202, 192, 314, 232, 367, 402, 418, 597, 483, 725, 772, 862, 1115, 966, 1408, 1445, 1690, 2066, 1869, 2672, 2663, 3209, 3777
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\12, x^(12*k)/prod(j=1, k, 1 - x^(5*j) + O(x*x^(n-12*k)))^4)/prod(j=0, n\5, 1 - x^(5*j+3) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(12*k)/(Product_{j=1..k} 1 - x^(5*j))^3)/(Product_{j>=0} 1 - x^(5*j+3)). - Andrew Howroyd, Sep 16 2019

A046785 Number of partitions of n with equal nonzero number of parts congruent to each of 0, 1, 3 and 4 (mod 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 4, 1, 5, 1, 5, 11, 5, 16, 6, 17, 26, 17, 41, 21, 46, 57, 47, 92, 61, 108, 122, 114, 193, 151, 233, 254, 253, 385, 344, 472, 520, 527, 750, 724, 922, 1035, 1054, 1426, 1458, 1746, 2015, 2035, 2676, 2811, 3248, 3818
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\13, x^(13*k)/prod(j=1, k, 1 - x^(5*j) + O(x*x^(n-13*k)))^4)/prod(j=0, n\5, 1 - x^(5*j+2) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(13*k)/(Product_{j=1..k} 1 - x^(5*j))^3)/(Product_{j>=0} 1 - x^(5*j+2)). - Andrew Howroyd, Sep 16 2019

A046786 Number of partitions of n with equal nonzero number of parts congruent to each of 0, 2, 3 and 4 (mod 5).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 5, 6, 6, 6, 6, 16, 21, 22, 22, 23, 43, 58, 63, 64, 68, 104, 139, 155, 160, 175, 236, 307, 347, 364, 405, 509, 640, 727, 772, 874, 1054, 1285, 1456, 1562, 1784, 2106, 2501, 2819, 3046, 3502, 4084, 4756, 5322, 5782
Offset: 0

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A046787.

Programs

  • PARI
    seq(n)={Vec(sum(k=1, n\14, x^(14*k)/prod(j=1, k, 1 - x^(5*j) + O(x*x^(n-14*k)))^4)/prod(j=0, n\5, 1 - x^(5*j+1) + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

G.f.: (Sum_{k>0} x^(14*k)/(Product_{j=1..k} 1 - x^(5*j))^3)/(Product_{j>=0} 1 - x^(5*j+1)). - Andrew Howroyd, Sep 16 2019
Showing 1-10 of 10 results.

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