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.

Previous Showing 21-30 of 46 results. Next

A338673 Expansion of Product_{k>=1} 1 / (1 - 4^(k-1)*x^k).

Original entry on oeis.org

1, 1, 5, 21, 101, 421, 2021, 8421, 39397, 167397, 766437, 3244517, 14881253, 62804453, 283415013, 1210159589, 5401907685, 22966866405, 102497423845, 435085808101, 1925197238757, 8215432696293, 36068400468453, 153579729097189, 674546796630501, 2866238341681637, 12508012102193637
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

Cf. A008284, A075900, A246936, A300579, A338674, A338675, A338676, A338677, A338678, A338679.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Product[1/(1 - 4^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 4^(n - k), {k, 0, n}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 4^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ sqrt(3) * polylog(2, 1/4)^(1/4) * 4^(n - 1/2) * exp(2*sqrt(polylog(2, 1/4)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338674 Expansion of Product_{k>=1} 1 / (1 - 5^(k-1)*x^k).

Original entry on oeis.org

1, 1, 6, 31, 181, 931, 5431, 27931, 159806, 834806, 4697306, 24478556, 137931681, 717306681, 3989650431, 20958791056, 115494337931, 604881056681, 3333662306681, 17439531447306, 95396181837931, 501716543166056, 2725636758009806, 14311071572462931, 77793648720900431
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

Cf. A008284, A075900, A246937, A300579, A338673, A338675, A338676, A338677, A338678, A338679.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Product[1/(1 - 5^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 5^(n - k), {k, 0, n}], {n, 0, 24}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 5^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 5^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ polylog(2, 1/5)^(1/4) * 5^(n - 1/2) * exp(2*sqrt(polylog(2, 1/5)*n)) / (sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338675 Expansion of Product_{k>=1} 1 / (1 - 6^(k-1)*x^k).

Original entry on oeis.org

1, 1, 7, 43, 295, 1807, 12391, 75895, 512647, 3179815, 21196807, 131258311, 875934727, 5416216711, 35763798535, 223059458311, 1461247179271, 9093600322567, 59586011601415, 370499158291975, 2411884242270727, 15072418547458567, 97530161503173127, 608700350537722375
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

Cf. A008284, A075900, A246938, A300579, A338673, A338674, A338676, A338677, A338678, A338679.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Product[1/(1 - 6^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 6^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 6^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 6^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ sqrt(5) * polylog(2, 1/6)^(1/4) * 6^(n - 1/2) * exp(2*sqrt(polylog(2, 1/6)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338676 Expansion of Product_{k>=1} 1 / (1 - 7^(k-1)*x^k).

Original entry on oeis.org

1, 1, 8, 57, 449, 3193, 25145, 178809, 1391314, 9996498, 76955586, 552257546, 4255024523, 30502987019, 232969386483, 1682476714724, 12762937304013, 92019035596293, 698222541789109, 5030814634614406, 37955614705675479, 274741644961416648, 2061916926761604144, 14909943849253537057
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

Cf. A008284, A075900, A246939, A300579, A338673, A338674, A338675, A338677, A338678, A338679.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Product[1/(1 - 7^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 7^(n - k), {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 7^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ sqrt(6) * polylog(2, 1/7)^(1/4) * 7^(n - 1/2) * exp(2*sqrt(polylog(2, 1/7)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338677 Expansion of Product_{k>=1} 1 / (1 - 8^(k-1)*x^k).

Original entry on oeis.org

1, 1, 9, 73, 649, 5257, 46729, 378505, 3331721, 27219593, 237491849, 1938544265, 16925054601, 138041874057, 1196384310921, 9820024329865, 84609648809609, 693596417152649, 5977550934234761, 48976660041553545, 419984680697190025, 3455551232025810569, 29494747047731910281
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

Cf. A008284, A075900, A246940, A300579, A338673, A338674, A338675, A338676, A338678, A338679.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - 8^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 8^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 8^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ sqrt(7) * polylog(2, 1/8)^(1/4) * 8^(n - 1/2) * exp(2*sqrt(polylog(2, 1/8)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338678 Expansion of Product_{k>=1} 1 / (1 - 9^(k-1)*x^k).

Original entry on oeis.org

1, 1, 10, 91, 901, 8191, 81091, 737191, 7239142, 66288142, 646149322, 5912729632, 57664985653, 527352541453, 5111015223223, 46998961540624, 453182267869615, 4163124744738505, 40151590267580785, 368699990679135946, 3540322181970716707, 32632895079429817528, 312061810101214595698
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Crossrefs

Cf. A008284, A075900, A246941, A300579, A338673, A338674, A338675, A338676, A338677, A338679.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - 9^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 9^(n - k), {k, 0, n}], {n, 0, 22}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 9^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ sqrt(2) * polylog(2, 1/9)^(1/4) * 9^(n - 1/2) * exp(2*sqrt(polylog(2, 1/9)*n)) / (sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A338679 Expansion of Product_{k>=1} 1 / (1 - 10^(k-1)*x^k).

Original entry on oeis.org

1, 1, 11, 111, 1211, 12211, 133211, 1343211, 14553211, 147653211, 1589753211, 16120753211, 173641753211, 1759951753211, 18855161753211, 192028261753211, 2048080361753211, 20841811361753211, 222333332361753211, 2261780642361753211, 24033895852361753211, 245331468952361753211
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 23 2021

Keywords

Comments

In general, if g.f. = Product_{k>=1} 1/(1 - d^(k-1)*x^k), where d > 1, then a(n) ~ sqrt(d-1) * polylog(2, 1/d)^(1/4) * d^(n - 1/2) * exp(2*sqrt(polylog(2, 1/d)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

Crossrefs

Cf. A008284, A075900, A246942, A300579, A338673, A338674, A338675, A338676, A338677, A338678.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Product[1/(1 - 10^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[Length[IntegerPartitions[n, {k}]] 10^(n - k), {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 10^(k - k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

Formula

a(n) = Sum_{k=0..n} p(n,k) * 10^(n-k), where p(n,k) = number of partitions of n into k parts.
a(n) ~ 3 * polylog(2, 1/10)^(1/4) *10^(n - 1/2) * exp(2*sqrt(polylog(2, 1/10)*n)) / (2*sqrt(Pi)*n^(3/4)). - Vaclav Kotesovec, May 09 2021

A358907 Number of finite sequences of distinct integer compositions with total sum n.

Original entry on oeis.org

1, 1, 2, 8, 18, 54, 156, 412, 1168, 3200, 8848, 24192, 66632, 181912, 495536, 1354880, 3680352, 9997056, 27093216, 73376512, 198355840, 535319168, 1443042688, 3884515008, 10445579840, 28046885824, 75225974912, 201536064896, 539339293824, 1441781213952
Offset: 0

Views

Author

Gus Wiseman, Dec 07 2022

Keywords

Examples

			The a(1) = 1 through a(4) = 18 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((12))     ((13))
                 ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((112))
                 ((2)(1))   ((121))
                 ((1)(11))  ((211))
                 ((11)(1))  ((1111))
                            ((1)(3))
                            ((3)(1))
                            ((1)(12))
                            ((11)(2))
                            ((1)(21))
                            ((12)(1))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))

Links

Crossrefs

For sets instead of sequences we have A098407, partitions A261049.
This is the strict case of A133494.
The case of distinct sums is A336127, constant sums A074854.
The version for sequences of partitions is A358906.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A218482 counts sequences of compositions with weakly decreasing lengths.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all different Omegas.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.

Programs

  • Maple
    g:= proc(n) option remember; ceil(2^(n-1)) end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
      add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 15 2022
  • Mathematica
    comps[n_]:=Join@@Permutations/@IntegerPartitions[n];
Table[Length[Select[Join@@Table[Tuples[comps/@c],{c,comps[n]}],UnsameQ@@#&]],{n,0,10}]

Extensions

a(16)-a(29) from Alois P. Heinz, Dec 15 2022

A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 26, 39, 78, 142, 320, 488, 913, 1558, 2798, 5865, 9482, 16742, 28474, 50814, 82800, 172540, 266093, 472432, 790824, 1361460, 2251665, 3844412, 7205416, 11370048, 19483502, 32416924, 54367066, 88708832, 149179800, 239738369, 445689392
Offset: 0

Views

Author

Gus Wiseman, Jul 19 2020

Keywords

Comments

A strict composition of n (A032020) is a finite sequence of distinct positive integers summing to n.
Is there a simple generating function?

Examples

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (2,1)    (3,1)      (3,2)
            (1),(2)  (1),(3)    (4,1)
            (2),(1)  (3),(1)    (1),(4)
                     (1),(2,1)  (2),(3)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1),(3,1)
                                (2,1),(2)
                                (2),(2,1)
                                (3,1),(1)

Links

Crossrefs

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of strict partitions are A072706.
Set partitions of strict partitions are A294617.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A318683, A318684, A319794, A323583, A336128, A336130, A336132, A336133.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

Programs

  • Mathematica
    strptn[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[strptn/@ctn],{ctn,Join@@Permutations/@strptn[n]}]],{n,0,10}]
  • PARI
    \\ here Q(N) gives A000009 as a vector.
Q(n) = {Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)))}
seq(n)={my(b=Q(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*b[1+k] + O(x*x^n)))]} \\ Andrew Howroyd, Apr 16 2021

Formula

G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000009(j)). - Andrew Howroyd, Apr 16 2021

A336141 Number of ways to choose a strict composition of each part of an integer partition of n.

Original entry on oeis.org

1, 1, 2, 5, 9, 17, 41, 71, 138, 270, 518, 938, 1863, 3323, 6163, 11436, 20883, 37413, 69257, 122784, 221873, 397258, 708142, 1249955, 2236499, 3917628, 6909676, 12130972, 21251742, 36973609, 64788378, 112103360, 194628113, 336713377, 581527210, 1000153063
Offset: 0

Views

Author

Gus Wiseman, Jul 18 2020

Keywords

Comments

A strict composition of n is a finite sequence of distinct positive integers summing to n.

Examples

			The a(1) = 1 through a(5) = 17 ways:
  (1)  (2)      (3)          (4)              (5)
       (1),(1)  (1,2)        (1,3)            (1,4)
                (2,1)        (3,1)            (2,3)
                (2),(1)      (2),(2)          (3,2)
                (1),(1),(1)  (3),(1)          (4,1)
                             (1,2),(1)        (3),(2)
                             (2,1),(1)        (4),(1)
                             (2),(1),(1)      (1,2),(2)
                             (1),(1),(1),(1)  (1,3),(1)
                                              (2,1),(2)
                                              (3,1),(1)
                                              (2),(2),(1)
                                              (3),(1),(1)
                                              (1,2),(1),(1)
                                              (2,1),(1),(1)
                                              (2),(1),(1),(1)
                                              (1),(1),(1),(1),(1)

Links

Crossrefs

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of partitions are A323583.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A316245, A318684, A319794, A336128, A336130, A336132, A336135.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 g(n$2):
seq(a(n), n=0..38);  # Alois P. Heinz, Jul 31 2020
  • Mathematica
    Table[Length[Join@@Table[Tuples[Join@@Permutations/@Select[IntegerPartitions[#],UnsameQ@@#&]&/@ctn],{ctn,IntegerPartitions[n]}]],{n,0,10}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0,
     If[n==0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p+1]]];
g[n_, i_] := g[n, i] = If[n==0 || i==1, 1, g[n, i-1] +
     b[i, i, 0] g[n-i, Min[n-i, i]]];
a[n_] := g[n, n];
a /@ Range[0, 38] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

Formula

G.f.: Product_{k >= 1} 1/(1 - A032020(k)*x^k).
Previous Showing 21-30 of 46 results. Next

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