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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A370715 a(n) = 3^(2*n) * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/3).

Original entry on oeis.org

1, 6, 126, 1818, 32130, 452142, 8006526, 117619290, 1999520154, 31550881374, 527781570174, 8556328428786, 145177242834330, 2404855490356782, 40907085509085750, 691705559193384114, 11840743106503713594, 202344257179543757526, 3487245860820904368822, 60077736592697832105330
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 27 2024

Keywords

Crossrefs

Cf. A070933, A370716.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[1/(1-2*x^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]
nmax = 25; CoefficientList[Series[Product[1/(1-2*(9*x)^k), {k, 1, nmax}]^(1/3), {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[(-1/QPochhammer[2,x])^(1/3), {x, 0, nmax}], x] * 9^Range[0, nmax]

Formula

G.f.: Product_{k>=1} 1/(1 - 2*(9*x)^k)^(1/3).
a(n) ~ c * 18^n / n^(2/3), where c = 1 / (Gamma(1/3) * QPochhammer(1/2)^(1/3)) = 0.564734286036917647642848904946237...

A264685 Expansion of Product_{k>=1} (1 + x^k)/(1 - 2*x^k).

Original entry on oeis.org

1, 3, 9, 24, 60, 141, 324, 717, 1560, 3330, 7020, 14622, 30225, 61998, 126522, 257007, 520326, 1050396, 2116116, 4255584, 8547330, 17149350, 34382295, 68889840, 137969466, 276220962, 552865365, 1106356314, 2213644548, 4428657402, 8859340926, 17721640698
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 21 2015

Keywords

Links

Crossrefs

Cf. A000009, A006951, A070933, A261584, A264686.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1 + x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 2^n, where c = A079555 / A048651 = Product_{k>=1} (2^k+1)/(2^k-1) = 8.25598793577825006554414084943227312652...

A269144 Expansion of Product_{k>=1} ((1 + k*x^k) / (1 - 2*x^k)).

Original entry on oeis.org

1, 3, 10, 29, 77, 195, 475, 1115, 2546, 5706, 12528, 27106, 57893, 122299, 255995, 531816, 1097377, 2252151, 4600835, 9362334, 18990645, 38418370, 77548880, 156251955, 314363615, 631703790, 1268148900, 2543812090, 5099469848, 10217529291, 20464112218
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 20 2016

Keywords

Comments

Convolution of A022629 and A070933.

Links

Crossrefs

Cf. A022629, A070933, A267004, A269153.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+k*x^k)/(1-2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 2^n, where c = Product_{k>=1} (2^k + k)/(2^k - 1) = 19.14883592186082265751161402244824703642181055238186925199088...

A269153 Expansion of Product_{k>=1} ((1 - k*x^k) / (1 - 2*x^k)).

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 33, 62, 130, 264, 554, 1081, 2237, 4483, 8952, 17933, 35921, 71755, 143502, 286713, 573198, 1146540, 2292277, 4584087, 9166802, 18334880, 36668210, 73336840, 146672469, 293348402, 586695560, 1173398119, 2346805311, 4693617598, 9387229673
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 20 2016

Keywords

Links

Crossrefs

Cf. A070933, A267005, A269144.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1-k*x^k)/(1-2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * 2^n, where c = Product_{k>=1} (2^k - k)/(2^k - 1) = 0.27320499481666294779155052256744055231134605935215258251663905...

A355389 Number of unordered pairs of distinct integer partitions of n.

Original entry on oeis.org

0, 0, 1, 3, 10, 21, 55, 105, 231, 435, 861, 1540, 2926, 5050, 9045, 15400, 26565, 43956, 73920, 119805, 196251, 313236, 501501, 786885, 1239525, 1915903, 2965830, 4528545, 6909903, 10417330, 15699606, 23403061, 34848726, 51435153, 75761895, 110744403, 161577276
Offset: 0

Views

Author

Gus Wiseman, Jul 04 2022

Keywords

Examples

			The a(0) = 0 through a(4) = 10 pairs:
  .  .  (2)(11)  (3)(21)    (4)(22)
                 (3)(111)   (4)(31)
                 (21)(111)  (22)(31)
                            (4)(211)
                            (22)(211)
                            (31)(211)
                            (4)(1111)
                            (22)(1111)
                            (31)(1111)
                            (211)(1111)

Crossrefs

The version for compositions is A006516.
Without distinctness we get A086737.
The unordered version is A355390, without distinctness A001255.
A000041 counts partitions, strict A000009.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
Cf. A022811, A070933, A316245, A317715, A319794, A319910, A323433, A339006, A355385.

Programs

  • Maple
    a:= n-> binomial(combinat[numbpart](n),2):
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 07 2024
  • Mathematica
    Table[Binomial[PartitionsP[n],2],{n,0,6}]
  • PARI
    a(n) = binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

Formula

a(n) = binomial(A000041(n), 2) = A355390(n)/2.

A265955 Expansion of Product_{k>=1} (1 + 2*k*x^k)/(1 - 2*k*x^k).

Original entry on oeis.org

1, 4, 16, 60, 192, 596, 1776, 5020, 13760, 36916, 96336, 246316, 619392, 1530548, 3729392, 8976364, 21337920, 50195268, 116977232, 270114764, 618712640, 1406843940, 3176387120, 7126185948, 15894370816, 35253947940, 77796242768, 170868178332, 373606888128
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 19 2015

Keywords

Links

Crossrefs

Cf. A006906, A022629, A032302, A032309, A070933, A265951.

Programs

  • Mathematica
    nmax=40; CoefficientList[Series[Product[(1+2*k*x^k)/(1-2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * n * 2^n, where c = 2 * Product_{m>=3} (1 + 2/(2^(m-1)/m - 1)) = 193.4198278838721371054040810054045645734538119720773785523616944906739...

A266943 Expansion of Product_{k>=1} 1 / (1 - 2*x^k)^2.

Original entry on oeis.org

1, 4, 16, 52, 160, 452, 1232, 3204, 8112, 19956, 48112, 113732, 264816, 607876, 1379264, 3096372, 6888784, 15201156, 33306752, 72510916, 156972960, 338089844, 724883552, 1547816708, 3292816416, 6981664708, 14758159472, 31110217524, 65415167744, 137230388228
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 06 2016

Keywords

Links

Crossrefs

Cf. A048651, A070933, A266944, A266945.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1-2*x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ c * n * 2^n, where c = 1/A048651^2 = 11.9906141505474711257730652493... .

A370732 a(n) = 4^n * [x^n] Product_{k>=1} 1/(1 - 2*x^k)^(1/4).

Original entry on oeis.org

1, 2, 18, 108, 822, 4796, 37492, 231704, 1738150, 11857004, 87262684, 617409128, 4638712124, 33724007896, 253800160808, 1894353653552, 14350905612038, 108412437326412, 827441075006796, 6308125533133896, 48388714839180756, 371391625244862600, 2860885559165073624
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Crossrefs

Cf. A070933 (m=1), A370713 (m=2), A370715 (m=3), A370733 (m=5).

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/(1-2*x^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x] * 4^Range[0, nmax]
nmax = 30; CoefficientList[Series[Product[1/(1-2*(4*x)^k), {k, 1, nmax}]^(1/4), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - 2*(4*x)^k)^(1/4).
a(n) ~ 8^n / (Gamma(1/4) * QPochhammer(1/2)^(1/4) * n^(3/4)).

A370735 a(n) = 5^(2*n) * [x^n] Product_{k>=1} 1/(1 - 3*x^k)^(1/5).

Original entry on oeis.org

1, 15, 1050, 52125, 3277500, 179801250, 11966690625, 738318187500, 49788716718750, 3314446448437500, 227432073022265625, 15631633385109375000, 1090877899335878906250, 76338563689129101562500, 5384934139819611328125000, 381204340327212964599609375, 27111589537137988341064453125
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 28 2024

Keywords

Comments

In general, if d > 1, m >= 1 and g.f. = Product_{k>=1} 1/(1 - d*x^k)^(1/m), then a(n) ~ d^n / (Gamma(1/m) * QPochhammer(1/d)^(1/m) * n^(1 - 1/m)).

Crossrefs

Cf. A242587 (d=3,m=1), A370714 (d=3,m=2), A370710 (d=3,m=3), A370734 (d=3,m=4).
Cf. A070933 (d=2,m=1), A370713 (d=2,m=2), A370715 (d=2,m=3), A370732 (d=2,m=4), A370733 (d=2,m=5).
Cf. A000041 (d=1,m=1), A271235 (d=1,m=2), A271236 (d=1,m=3).

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1-3*x^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x] * 25^Range[0, nmax]
nmax = 20; CoefficientList[Series[Product[1/(1-3*(25*x)^k), {k, 1, nmax}]^(1/5), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - 3*(25*x)^k)^(1/5).
a(n) ~ 75^n / (Gamma(1/5) * QPochhammer(1/3)^(1/5) * n^(4/5)).

A262444 Number of 3-colored integer partitions such that no adjacent parts have the same color.

Original entry on oeis.org

1, 3, 9, 21, 51, 111, 249, 525, 1119, 2319, 4809, 9825, 20079, 40671, 82341, 165945, 334191, 671307, 1347861, 2702385, 5416395, 10847787, 21720981, 43474869, 87004875, 174081051, 348279777, 696712749, 1393674603, 2787673767, 5575871457, 11152425093, 22305942039
Offset: 0

Views

Author

Ran Pan, Sep 23 2015

Keywords

Examples

			a(2) = 9 because there are two integer partitions of 2: [2], [1,1] and there are three ways to color [2] and 3 X 2 = 6 ways to color [1,1].

Links

Crossrefs

Cf. A070933, A262495.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> floor(b(n$2)/2*3):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 23 2015
  • Mathematica
    Rest[CoefficientList[Series[3/2 Product[1/(1 - 2 x^k), {k, 1, 35}], {x, 0, 35}], x]] (* Vincenzo Librandi, Sep 23 2015 *)

Formula

G.f.: -1/2 + (3/2)*Product_{k>=1} 1/(1-2*x^k).
a(n) = floor(3/2*A070933(n)).
a(n) = Sum_{k=0..3} 6/k! * A262495(n,3-k). - Alois P. Heinz, Sep 24 2015
Previous Showing 21-30 of 45 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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