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 1-6 of 6 results.

A286955 n-vertex sequences of plane forests with nondecreasing numbers of trees.

Original entry on oeis.org

1, 1, 3, 9, 29, 96, 326, 1127, 3952, 14019, 50208, 181275, 659039, 2410433, 8862750, 32739168, 121443136, 452167865, 1689237104, 6330103627, 23787215202, 89616350271, 338417312294, 1280739676563, 4856711761475, 18451630811041, 70223495698892, 267691953822783
Offset: 0

Views

Author

David Bevan, May 22 2017

Keywords

Comments

Enumerates Part[Cat], the substitution of Cat for atoms of Part, where Part is the set of integer partitions (A000041), and Cat is any set counted by the 1-based Catalan numbers (A000108 shifted).

Examples

			a(3) = 9, consisting of (1,1,1), (1,2), (2,1), (3a), (3b), (1)(1,1), (1)(2), (2)(1), and (1)(1)(1), where 1 is the one-vertex tree, 2 is the two-vertex tree, 3a and 3b are the two three-vertex trees, and parentheses record the partitioning into forests. (1,1)(1) is excluded because the numbers of trees per forest decreases.

Links

Crossrefs

Cf. A000041, A000108, A307496.

Programs

  • Mathematica
    m = 20; CoefficientList[Series[Product[1/(1-((1-Sqrt[1-4x])/2)^k),{k,m}],{x,0,m}],x]
nmax = 30; CoefficientList[Series[1/QPochhammer[(1 - Sqrt[1 - 4*x])/2], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 10 2020 *)
Join[{1}, Table[Sum[(k/(2*n - k))*Binomial[2*n - k, n - k]*PartitionsP[k], {k, 0, n}], {n, 1, 30}]] (* Vaclav Kotesovec, Jul 31 2022 *)

Formula

G.f.: Product_{k>0} 1/(1 - ((1 - sqrt(1 - 4*x))/2)^k), the composition of the g.f. for A000041 with x times the g.f. for A000108.
a(n) ~ c * 4^n / n^(3/2), where c = 1/sqrt(Pi) * Sum_{k>=0} k*A000041(k)/2^(k+1) = 2.680434829690402658212615372294526133126515771886321123341424399596963885434... - Vaclav Kotesovec, Jun 02 2018, extended Aug 01 2022

A307501 Expansion of Product_{k>=1} (1 + (x*(1 - x))^k).

Original entry on oeis.org

1, 1, 0, 0, -3, 1, -1, 3, 3, 0, -12, 15, -20, 5, 53, -113, 180, -241, 153, 173, -652, 787, 628, -4801, 11635, -18699, 20775, -12315, -6109, 21253, -7015, -61060, 174382, -260676, 190623, 130141, -549572, 399845, 1577502, -6670524, 14603574, -21111528, 16110192, 14794188, -82586174
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 11 2019

Keywords

Links

Crossrefs

Cf. A000009, A030528, A129519, A266108, A307496, A307500.

Programs

  • Mathematica
    nmax = 44; CoefficientList[Series[Product[(1 + (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] (x (1 - x))^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsQ[k], {k, 0, n}], {n, 0, 44}]

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d ) * (x*(1 - x))^k/k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000009(k).

A356268 a(n) = Sum_{k=0..n} binomial(2*n, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 3, 11, 62, 289, 1472, 7581, 38014, 184453, 918512, 4548393, 22077762, 107423503, 516720332, 2483445404, 11959145079, 57022343425, 270173627092, 1282971321633, 6047971597490, 28446033085527, 133714464665108, 625893086713686, 2919093380089383, 13596052503945537
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

Crossrefs

Cf. A000009, A032443, A266232, A307496, A356267.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]

Formula

a(n) ~ erfc(Pi/(4*sqrt(3))) * 2^(2*n - 3) * exp(Pi*sqrt(n/3) + Pi^2/48) / (3^(1/4) * n^(3/4)).

A356281 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 3, 11, 43, 172, 695, 2823, 11501, 46940, 191791, 784148, 3207196, 13119733, 53670793, 219545353, 897957702, 3672093558, 15013596535, 61370565546, 250803861369, 1024716136043, 4185683293934, 17093143284723, 69786349712519, 284847779542644, 1162385753008079
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

Crossrefs

Cf. A000009, A032443, A266232, A307496, A356268, A356280.

Programs

  • Mathematica
    Table[Sum[PartitionsQ[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 2^(2*n - 1/2) * exp(3^(1/3) * Pi^(4/3) * n^(1/3) / 2^(8/3)) / sqrt(3*Pi*n).

A356270 a(n) = Sum_{k=0..n} binomial(2*k, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 3, 9, 49, 189, 945, 4641, 21801, 99021, 487981, 2335541, 10800725, 51363065, 238573865, 1121139065, 5309312105, 24543884585, 113220920945, 530677144745, 2439321389945, 11261499234425, 52169097691865, 239433905462945, 1095710701133345, 5029918350471545
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

Crossrefs

Cf. A000009, A006134, A032443, A266232, A307496, A356268, A356269.

Programs

  • Mathematica
    Table[Sum[Binomial[2*k, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]

Formula

a(n) ~ binomial(2*n,n) * q(n) * 4/3.
a(n) ~ 2^(2*n) * exp(Pi*sqrt(n/3)) / (3^(5/4) * sqrt(Pi) * n^(5/4)).

A307495 Expansion of Sum_{k>=0} k!*((1 - sqrt(1 - 4*x))/2)^k.

Original entry on oeis.org

1, 1, 3, 12, 57, 312, 1950, 13848, 111069, 998064, 9957186, 109305240, 1309637274, 17006109072, 237888664572, 3566114897520, 57030565449765, 969154436550240, 17439499379433690, 331268545604793240, 6624013560942038670, 139080391965533653200, 3059323407592802838180, 70355685298375014175440
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2019

Keywords

Comments

Catalan transform of A000142 (factorial numbers).
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by putting the sequence of factorial numbers in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.
1
1 1
2 3 3
6 9 12 12
24 33 45 57 57
120 153 198 255 312 312
...
Alternatively, the sequence can be obtained by multiplying the sequence of factorial numbers by the array A106566.
(End)

Links

Crossrefs

Cf. A000108, A000142, A013999, A100100, A106566, A307496.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Sum[k! ((1 - Sqrt[1 - 4 x])/2)^k, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] (1 - Sqrt[1 - 4 x])/2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
Join[{1}, Table[1/n Sum[Binomial[2n - k - 1, n - k] k k!, {k, n}], {n, 23}]]

Formula

G.f.: 1 /(1 - x*c(x)/(1 - x*c(x)/(1 - 2*x*c(x)/(1 - 2*x*c(x)/(1 - 3*x*c(x)/(1 - 3*x*c(x)/(1 - ...))))))), a continued fraction, where c(x) = g.f. of Catalan numbers (A000108).
Sum_{n>=0} a(n)*(x*(1 - x))^n = g.f. of A000142.
a(n) = (1/n) * Sum_{k=1..n} binomial(2*n-k-1,n-k)*k*k! for n > 0.
a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Aug 10 2019
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.