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.

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A332199 Expansion of Product_{i>=1, j>=1} 1/(1 - i*x^(i*j)).

Original entry on oeis.org

1, 1, 4, 8, 22, 40, 101, 183, 412, 765, 1586, 2899, 5834, 10484, 20199, 36246, 67758, 119837, 219661, 384200, 690164, 1197423, 2114105, 3632088, 6332797, 10779478, 18555115, 31354932, 53385037, 89494901, 150983344, 251284829, 420218575, 694947117, 1152915743, 1894656801
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2020

Keywords

Links

Crossrefs

Cf. A006171, A006906, A285245, A318415, A333653.

Programs

  • Mathematica
    m = 35; CoefficientList[Series[Product[1/(1 - i*x^(i*j)), {i, 1, m}, {j, 1, m}], {x, 0, m}], x] (* Amiram Eldar, Aug 23 2020 *)
  • PARI
    N=40; x='x+O('x^N); Vec(1/prod(i=1, N, prod(j=1, N\i, 1-i*x^(i*j))))
  • PARI
    N=40; x='x+O('x^N); Vec(1/prod(k=1, N, prod(d=1, k, 1-(k%d==0)*d*x^k)))

Formula

G.f.: Product_{k>0} f(q^k) where f(q) = Product_{i>=1} 1/(1 - i*q^i).
G.f.: Product_{k>0} Product_{d|k} 1/(1 - d*x^k).

A344368 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(1-s)).

Original entry on oeis.org

1, 2, 3, 8, 5, 12, 7, 24, 18, 20, 11, 48, 13, 28, 30, 80, 17, 72, 19, 80, 42, 44, 23, 168, 50, 52, 81, 112, 29, 150, 31, 224, 66, 68, 70, 324, 37, 76, 78, 280, 41, 210, 43, 176, 180, 92, 47, 576, 98, 200, 102, 208, 53, 378, 110, 392, 114, 116, 59, 660
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2021

Keywords

Crossrefs

Cf. A001055, A006906, A050367, A050369, A224892, A344369, A344370.

Programs

  • Mathematica
    T[, 1] = T[1, ] = 1; T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] T[n/#, #] &]; A001055[n_] := T[n, n]; Table[n A001055[n], {n, 60}]

Formula

a(n) = n * A001055(n).
a(1) = 1; a(n) = -Sum_{d|n, d < n} A224892(n/d) * a(d).

A387179 Number of twice-partitions of n into distinct constant partitions.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 18, 28, 48, 69, 105, 158, 240, 343, 503, 720, 1041, 1459, 2062, 2874, 4047, 5547, 7656, 10472, 14322, 19360, 26214, 35192, 47354, 63030, 83992, 111258, 147360, 193804, 254907, 333553, 436319, 567673, 738197, 956049, 1237453, 1594700, 2053361
Offset: 0

Views

Author

Gus Wiseman, Sep 07 2025

Keywords

Comments

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

Examples

			The a(1) = 1 through a(6) = 18 twice-partitions counted by this sequence:
  (1)  (2)   (3)      (4)       (5)         (6)
       (11)  (111)    (22)      (3)(2)      (33)
             (2)(1)   (1111)    (4)(1)      (222)
             (11)(1)  (3)(1)    (11111)     (4)(2)
                      (11)(2)   (22)(1)     (5)(1)
                      (2)(11)   (3)(11)     (22)(2)
                      (111)(1)  (111)(2)    (4)(11)
                                (111)(11)   (111111)
                                (1111)(1)   (111)(3)
                                (11)(2)(1)  (22)(11)
                                (2)(11)(1)  (3)(111)
                                            (1111)(2)
                                            (3)(2)(1)
                                            (1111)(11)
                                            (11111)(1)
                                            (3)(11)(1)
                                            (111)(2)(1)
                                            (111)(11)(1)

Links

Crossrefs

The non-distinct version is A279784.
Dominates the case of distinct block-sums A279786.
This is the constant-block case of A296122.
For strict instead of constant partitions we have A358914.
A000041 counts integer partitions, strict A000009.
A047968 counts constant twice-partitions.
A063834 counts twice-partitions.
Cf. A387120, zeros A387180 (counted by A387329), nonzeros A387181 (counted by A387330).
Cf. A000005, A006906, A007425, A261049, A270995, A299200, A299201, A355739, A387110.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
      binomial(numtheory[tau](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 08 2025
  • Mathematica
    twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@#&&And@@SameQ@@@#&]],{n,0,10}]

Extensions

More terms from Alois P. Heinz, Sep 08 2025

A182779 a(n) = A049019(n) * A118851(n). Irregular table read by rows.

Original entry on oeis.org

1, 1, 2, 2, 3, 12, 6, 4, 24, 24, 72, 24, 5, 40, 120, 180, 360, 480, 120, 6, 60, 240, 180, 360, 2160, 720, 1440, 4320, 3600, 720, 7, 84, 420, 840, 630, 5040, 3780, 7560, 3360, 30240, 20160, 12600, 50400, 30240, 5040
Offset: 0

Views

Author

Alford Arnold, Dec 01 2010

Keywords

Comments

The sequences have shape A000041 and their respective row sums are A000670, A006906 and A006153.

Examples

			For n = 3 the values are (3,12,6) = (1,6,6)*(3,2,1).
Table starts:
1;
1;
2, 2;
3, 12, 6;
4, (24, 24), 72, 24;
5, (40, 120), (180, 360), 480, 120;
6, (60, 240, 180), (360, 2160, 720), (1440, 4320), 3600, 720;
7, (84, 420, 840), (630, 5040, 3780, 7560), (3360, 30240, 20160), (12600, 50400), 30240, 5040;

Crossrefs

Cf. A006153 (related to function composition), A133314 (signed version of A049019).

Formula

a(n) = A049019(n) * A118851(n).

Extensions

a(0) = 1 prepended by Peter Luschny, May 31 2020

A282209 Expansion of Product_{k>=1} 1/(1 - k^2*x^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 5, 21, 30, 30, 30, 94, 130, 130, 130, 402, 546, 627, 627, 1715, 2291, 2615, 2615, 6967, 9440, 10736, 11465, 28873, 38765, 43949, 46865, 116753, 156321, 178578, 190242, 476391, 634663, 723691, 770347, 1914943, 2550735, 2906847, 3107160, 7685544
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 09 2017

Keywords

Comments

Sum of products of terms in all partitions of n into squares (A000290).

Examples

			a(8) = 21 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1], 4*4 = 16, 4*1*1*1*1 = 4, 1*1*1*1*1*1*1*1 = 1 and 16 + 4 + 1 = 21.

Links

Crossrefs

Cf. A000290, A001156, A006906.

Programs

  • Mathematica
    nmax = 44; CoefficientList[Series[Product[1/(1 - k^2 x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - k^2*x^(k^2)).
From Vaclav Kotesovec, Feb 09 2017: (Start)
a(n) ~ c * 2^(n/2), where:
c = 1.84902025727376837058629436557644856279088... if n == 0 (mod 4),
c = 1.74739571210218418633067606853005648684028... if n == 1 (mod 4),
c = 1.41060067910504703778072732362810764186990... if n == 2 (mod 4),
c = 1.06705333199321743850009229910087278853310... if n == 3 (mod 4).
(End)

A302288 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - k*x^k*A(x)).

Original entry on oeis.org

1, 1, 4, 14, 55, 217, 908, 3864, 16894, 75078, 338862, 1548055, 7147427, 33294790, 156305144, 738753341, 3512431392, 16788169689, 80619590577, 388785776751, 1882063496033, 9142361671588, 44550166132194, 217716111661799, 1066792279046783, 5239947708977474, 25795965431819883
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 04 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 4*x^2 + 14*x^3 + 55*x^4 + 217*x^5 + 908*x^6 + 3864*x^7 + 16894*x^8 + 75078*x^9 + 338862*x^10 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - 2*x^2*A(x)) * (1 - 3*x^3*A(x)) * (1 - 4*x^4*A(x)) * ...).

Crossrefs

Cf. A006906, A145268, A298261, A301577, A302171, A302289.

A307258 Expansion of (1/(1 + x)) * Product_{k>=1} 1/(1 - k*x^k/(1 + x)^k).

Original entry on oeis.org

1, 0, 2, -1, 5, -11, 36, -107, 311, -850, 2208, -5519, 13566, -33562, 84937, -220307, 579413, -1522616, 3954016, -10100863, 25416877, -63324271, 157248035, -391478354, 980410093, -2470810086, 6253495883, -15846525758, 40093721908, -101116823798, 254093749587, -636547773777
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

Inverse binomial transform of A006906.

Crossrefs

Cf. A006906, A281425, A294501, A307260, A318127.

Programs

  • Maple
    a:=series((1/(1+x))*mul(1/(1-k*x^k/(1+x)^k),k=1..100),x=0,32): seq(coeff(a,x,n),n=0..31); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 31; CoefficientList[Series[1/(1 + x) Product[1/(1 - k x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 31}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A006906(k).

A307261 Expansion of Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 1, 4, 13, 42, 130, 397, 1197, 3566, 10517, 30760, 89293, 257397, 737220, 2099215, 5945594, 16756258, 47004829, 131286914, 365203797, 1012031772, 2794446326, 7690009600, 21094325177, 57687762889, 157306741287, 427777384499, 1160250104637, 3139067594584, 8472525405830, 22815639395641
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

First differences of the binomial transform of A006906.

Crossrefs

Cf. A006906, A218482, A307262, A318127, A320563.

Programs

  • Maple
    a:=series(mul(1/(1-k*x^k/(1-x)^k),k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1 - k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

A307494 Expansion of Product_{j>=1} 1/(1 - j*(-1 + Product_{k>=1} 1/(1 - k*x^k))^j).

Original entry on oeis.org

1, 1, 6, 30, 145, 680, 3151, 14394, 65217, 293223, 1310255, 5820697, 25725139, 113161286, 495659656, 2162471602, 9399682398, 40716499477, 175798072996, 756709512011, 3247830724594, 13901967775738, 59352638426839, 252778786749676, 1074061758972744, 4553583433874616, 19264461634793094
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 15 2019

Keywords

Crossrefs

Cf. A006906, A307127, A307321, A307566.

Programs

  • Mathematica
    nmax = 26; CoefficientList[Series[Product[1/(1 - j (-1 + Product[1/(1 - k x^k), {k, 1, nmax}])^j), {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: g(g(x) - 1), where g(x) = g.f. of A006906.

A318025 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} 1/(1 - j*x^(k*j))).

Original entry on oeis.org

1, 4, 7, 18, 26, 66, 98, 216, 361, 701, 1171, 2287, 3763, 6887, 11707, 20740, 34637, 60678, 100581, 172609, 285924, 481671, 791317, 1323831, 2156856, 3561119, 5784021, 9459559, 15250217, 24783964, 39713789, 64032664, 102200203, 163617694, 259745174, 413886941, 653715969, 1035539948
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 23 2018

Keywords

Comments

Inverse Moebius transform of A006906.

Links

Crossrefs

Cf. A006906, A047968, A300277, A302549, A318290.

Programs

  • Mathematica
    nmax = 38; Rest[CoefficientList[Series[Sum[-1 + Product[1/(1 - j x^(k j)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]]
b[n_] := b[n] = SeriesCoefficient[Product[1/(1 - k x^k), {k, 1, n}], {x, 0, n}]; a[n_] := a[n] = SeriesCoefficient[Sum[b[k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}]; Table[a[n], {n, 38}]
Table[Sum[Total[Times @@@ IntegerPartitions[d]], {d, Divisors[n]}], {n, 38}]

Formula

G.f.: Sum_{k>=1} A006906(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} A006906(d).
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