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 1-9 of 9 results.

A304967 Expansion of Product_{k>=1} 1/(1 - x^k)^(p(k)-p(k-1)), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 9, 20, 26, 49, 68, 123, 173, 295, 432, 707, 1044, 1672, 2483, 3900, 5817, 8993, 13424, 20539, 30609, 46399, 69052, 103879, 154198, 230550, 341261, 507484, 749028, 1108559, 1631340, 2404311, 3527615, 5179317, 7577263, 11086413, 16173577, 23588227
Offset: 0

Views

Author

Ilya Gutkovskiy, May 22 2018

Keywords

Comments

Euler transform of A002865.

Links

Crossrefs

Cf. A000041, A000219, A001383, A001970, A002865, A304966.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)-1)*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 22 2018
  • Mathematica
    nmax = 42; CoefficientList[Series[Product[1/(1 - x^k)^(PartitionsP[k] - PartitionsP[k - 1]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (PartitionsP[d] - PartitionsP[d - 1]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 42}]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^A002865(k).

A320289 Number of phylogenetic trees with n labels and no singleton leaves.

Original entry on oeis.org

0, 1, 1, 4, 11, 86, 477, 4810, 40679, 496522, 5662933, 81759910, 1169640551, 19622623190, 336215135973, 6455705990674, 128445712218263, 2785761076726066, 62980942321570981, 1525318051255683598, 38566041706375722071, 1032726237783455193662
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Examples

			The a(2) = 1 through a(5) = 11 phylogenetic trees:
  (12)  (123)  (1234)      (12345)
               ((12)(34))  ((12)(345))
               ((13)(24))  ((13)(245))
               ((14)(23))  ((14)(235))
                           ((15)(234))
                           ((23)(145))
                           ((24)(135))
                           ((25)(134))
                           ((34)(125))
                           ((35)(124))
                           ((45)(123))

Links

Crossrefs

Cf. A000311, A000669, A002865, A005804, A141268, A300660, A304966, A304967, A319312, A320294, A320295.

Programs

  • Mathematica
    numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
rotf[n_]:=rotf[n]=If[n==1,0,1+Sum[numSetPtnsOfType[p]*Times@@rotf/@p,{p,Select[IntegerPartitions[n],Length[#]>1&]}]];
Array[rotf,20]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(v=vector(n)); for(n=2, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); v}
seq(n)={my(M=Mat(vectorv(n, k, b(n,k)))); vector(n, k, sum(i=1, k, binomial(k, i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

A320294 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n with no 1's.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 37, 48, 87, 126, 227, 342, 611, 964, 1719, 2806, 4975, 8327, 14782, 25157, 44609, 76972, 136622, 237987, 422881, 742149, 1320825, 2331491, 4156392, 7370868, 13164429, 23433637, 41928557, 74871434, 134203411, 240284935, 431437069
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Comments

Also phylogenetic trees with no singleton leaves on integer partitions of n with no 1's.

Examples

			The a(4) = 1 through a(10) = 15 trees:
  (22)  (32)  (33)   (43)   (44)        (54)        (55)
              (42)   (52)   (53)        (63)        (64)
              (222)  (322)  (62)        (72)        (73)
                            (332)       (333)       (82)
                            (422)       (432)       (433)
                            (2222)      (522)       (442)
                            ((22)(22))  (3222)      (532)
                                        ((22)(23))  (622)
                                                    (3322)
                                                    (4222)
                                                    (22222)
                                                    ((22)(24))
                                                    ((22)(33))
                                                    ((23)(23))
                                                    ((22)(222))

Links

Crossrefs

Cf. A000045, A000311, A000669, A002865, A141268, A292504, A304966, A304967, A319312, A320289, A320293, A320295, A320296.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,10}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=2, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=2, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Extensions

Terms a(16) and beyond from Andrew Howroyd, Oct 25 2018

A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

0, 1, 2, 5, 8, 19, 34, 80, 165, 394, 892, 2192, 5232, 13057, 32271, 81568, 205748, 525735, 1344828, 3467415, 8960849, 23280323, 60639680, 158559047, 415631368, 1092734050, 2879420753, 7605713020, 20130266302, 53386744298, 141836904569, 377479973474, 1006189769886
Offset: 1

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Comments

Also phylogenetic trees with no singleton leaves on integer partitions of n.

Examples

			The a(2) = 1 through a(6) = 19 trees:
  (11)  (21)   (22)        (32)         (33)
        (111)  (31)        (41)         (42)
               (211)       (221)        (51)
               (1111)      (311)        (222)
               ((11)(11))  (2111)       (321)
                           (11111)      (411)
                           ((11)(12))   (2211)
                           ((11)(111))  (3111)
                                        (21111)
                                        (111111)
                                        ((11)(13))
                                        ((11)(22))
                                        ((12)(12))
                                        ((11)(112))
                                        ((12)(111))
                                        ((11)(1111))
                                        ((111)(111))
                                        ((11)(11)(11))
                                        ((11)((11)(11)))

Links

Crossrefs

Cf. A000311, A000669, A005804, A141268, A302545, A304966, A319312, A320289, A320294.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,IntegerPartitions[n]}],{n,14}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Extensions

Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018

A305651 Expansion of Product_{k>=1} (1 + x^k)^(q(k)-1), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 5, 7, 12, 17, 26, 39, 59, 87, 132, 192, 284, 419, 612, 892, 1303, 1887, 2730, 3945, 5677, 8154, 11689, 16711, 23839, 33960, 48244, 68432, 96888, 136922, 193148, 272058, 382508, 537007, 752735, 1053550, 1472406, 2054988, 2863993, 3986245, 5541008
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 07 2018

Keywords

Comments

Weigh transform of A111133.
Convolution of the sequences A050342 and A081362.

Links

Crossrefs

Cf. A000009, A050342, A081362, A089259, A111133, A304966, A304969.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i)-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018
  • Mathematica
    nmax = 45; CoefficientList[Series[Product[(1 + x^k)^(PartitionsQ[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[(-1)^(k + 1)/k (1/ QPochhammer[x^k, x^(2 k)] - 1/(1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (PartitionsQ[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

Formula

G.f.: Product_{k>=1} (1 + x^k)^A111133(k).
G.f.: Product_{k>=1} (1 + x^k)^(A000009(k)-1).

A317535 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} 1/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 23, 48, 106, 227, 494, 1065, 2310, 4991, 10808, 23376, 50593, 109455, 236858, 512479, 1108924, 2399418, 5191853, 11233929, 24307777, 52596430, 113806948, 246252376, 532834797, 1152933975, 2494689316, 5397944266, 11679933875, 25272740480, 54684508281, 118324934647
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 30 2018

Keywords

Comments

Invert transform of A000065.

Links

Crossrefs

Cf. A000041, A000065, A001970, A055887, A063834, A261049, A271619, A304966, A317536.

Programs

  • Maple
    seq(coeff(series(1/(1+1/(1-x)-mul(1/(1-x^k),k=1..n)), x,n+1),x,n),n=0..40); # Muniru A Asiru, Jul 30 2018
  • Mathematica
    nmax = 35; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/(1 - Sum[(PartitionsP[k] - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(PartitionsP[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 35}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A000065(k)*x^k).

A358828 Number of twice-partitions of n with no singletons.

Original entry on oeis.org

1, 0, 1, 2, 5, 8, 19, 30, 68, 111, 229, 380, 799, 1280, 2519, 4325, 8128, 13666, 25758, 43085, 79300, 134571, 240124, 407794, 730398, 1224821, 2152122, 3646566, 6338691, 10657427, 18469865, 30913539, 53108364, 88953395, 151396452, 253098400, 429416589
Offset: 0

Views

Author

Gus Wiseman, Dec 03 2022

Keywords

Comments

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

Examples

			The a(2) = 1 through a(6) = 19 twice-partitions:
  (11)  (21)   (22)      (32)       (33)
        (111)  (31)      (41)       (42)
               (211)     (221)      (51)
               (1111)    (311)      (222)
               (11)(11)  (2111)     (321)
                         (11111)    (411)
                         (21)(11)   (2211)
                         (111)(11)  (3111)
                                    (21111)
                                    (111111)
                                    (21)(21)
                                    (22)(11)
                                    (31)(11)
                                    (111)(21)
                                    (21)(111)
                                    (211)(11)
                                    (111)(111)
                                    (1111)(11)
                                    (11)(11)(11)

Crossrefs

The version for multiset partitions of integer partitions is A304966.
Allowing singletons other than (1) gives A358829.
A002865 counts partitions with no 1's.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000009, A000041, A000219, A001970, A072233, A358824.

Programs

  • Mathematica
    twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],FreeQ[Length/@#,1]&]],{n,0,10}]

Formula

G.f.: Product_{k>=1} 1/(1-(A000041(k)-1)*x^k).

A358829 Number of twice-partitions of n with no (1)'s.

Original entry on oeis.org

1, 0, 2, 3, 9, 13, 38, 56, 144, 237, 524, 886, 1961, 3225, 6700, 11702, 23007, 39787, 77647, 133707, 254896, 442736, 820703, 1427446, 2630008, 4535330, 8224819, 14250148, 25513615, 43981753, 78252954, 134323368, 236900355, 406174046, 709886932, 1213934012
Offset: 0

Views

Author

Gus Wiseman, Dec 03 2022

Keywords

Comments

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

Examples

			The a(2) = 2 through a(5) = 13 twice-partitions:
  (2)   (3)    (4)       (5)
  (11)  (21)   (22)      (32)
        (111)  (31)      (41)
               (211)     (221)
               (1111)    (311)
               (2)(2)    (2111)
               (11)(2)   (3)(2)
               (2)(11)   (11111)
               (11)(11)  (21)(2)
                         (3)(11)
                         (111)(2)
                         (21)(11)
                         (111)(11)

Crossrefs

The version for multiset partitions of integer partitions is A317911.
Forbidding all singletons gives A358828.
A002865 counts partitions with no 1's.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000009, A000041, A000219, A001970, A072233, A304966, A358824.

Programs

  • Mathematica
    twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],FreeQ[Total/@#,1]&]],{n,0,10}]

Formula

G.f.: Product_{k>=2} 1/(1-A000041(k)*x^k).

A320291 Number of singleton-free multiset partitions of integer partitions of n with no 1's.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 3, 3, 7, 8, 15, 19, 36, 46, 79, 110, 181, 254, 407, 580, 907, 1309, 2004, 2909, 4410, 6407, 9599, 13984, 20782, 30252, 44677, 64967, 95414, 138563, 202527, 293583, 427442, 618337, 897023, 1295020, 1872696, 2697777, 3889964, 5591917, 8041593, 11535890
Offset: 0

Views

Author

Gus Wiseman, Oct 09 2018

Keywords

Examples

			The a(4) = 1 through a(10) = 15 multiset partitions:
  ((22))  ((23))  ((24))   ((25))   ((26))      ((27))      ((28))
                  ((33))   ((34))   ((35))      ((36))      ((37))
                  ((222))  ((223))  ((44))      ((45))      ((46))
                                    ((224))     ((225))     ((55))
                                    ((233))     ((234))     ((226))
                                    ((2222))    ((333))     ((235))
                                    ((22)(22))  ((2223))    ((244))
                                                ((22)(23))  ((334))
                                                            ((2224))
                                                            ((2233))
                                                            ((22222))
                                                            ((22)(24))
                                                            ((22)(33))
                                                            ((23)(23))
                                                            ((22)(222))

Links

Crossrefs

Cf. A002865, A007716, A049311, A083751, A283877, A293606, A302545, A304966, A304967, A320294, A320295, A320296.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@Select[IntegerPartitions[n],FreeQ[#,1]&],FreeQ[Length/@#,1]&]],{n,20}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n,i,i>1)); concat([1], EulerT(EulerT(v)-v))} \\ Andrew Howroyd, Oct 25 2018

Formula

Euler transform of A083751. - Andrew Howroyd, Oct 25 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Oct 25 2018
Showing 1-9 of 9 results.

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