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 11-19 of 19 results.

A248890 Number of rooted trees with n nodes such that for each inner node no more than k subtrees corresponding to its children have exactly k nodes.

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 8, 16, 34, 75, 166, 374, 849, 1952, 4522, 10566, 24840, 58760, 139693, 333702, 800412, 1927207, 4655997, 11283835, 27423930, 66825194, 163227234, 399587270, 980222058, 2409181633, 5931839530, 14629639579, 36137308192, 89395224033
Offset: 0

Views

Author

Alois P. Heinz, Mar 05 2015

Keywords

Examples

			:  o  :  o  :  o  :    o   o  :    o     o     o   o  :
:     :  |  :  |  :   / \  |  :    |    / \   / \  |  :
:     :  o  :  o  :  o   o o  :    o   o   o o   o o  :
:     :     :  |  :  |     |  :   / \  |     |   | |  :
:     :     :  o  :  o     o  :  o   o o     o   o o  :
:     :     :     :        |  :  |     |           |  :
:     :     :     :        o  :  o     o           o  :
:     :     :     :           :                    |  :
: n=1 : n=2 : n=3 :  n=4      :  n=5               o  :
:.....:.....:.....:...........:.......................:

Links

Crossrefs

Cf. A000081, A032305, A045648, A213920.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(g((i-1)$2)+j-1, j)*g(n-i*j, i-1), j=0..min(i, n/i))))
    end:
a:= n-> g((n-1)$2):
seq(a(n), n=0..40);
  • Mathematica
    g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i-1, i-1]+j-1, j]*g[n-i*j, i-1], {j, 0, Min[i, n/i]}]]]; a[n_] := g[n-1, n-1]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 28 2017, translated from Maple *)

A307538 G.f. A(x) satisfies: A(x) = x*exp(2*A(-x) + 2*A(-x^3)/3 + 2*A(-x^5)/5 + 2*A(-x^7)/7 + 2*A(-x^9)/9 + ...).

Original entry on oeis.org

0, 1, -2, -2, 10, 14, -86, -126, 858, 1302, -9378, -14606, 108954, 172698, -1319966, -2119118, 16489594, 26731542, -210887998, -344490170, 2747510514, 4515757426, -36336187630, -60023827438, 486540793914, 807121753178, -6582918170714, -10959656342678, 89860260268098
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 14 2019

Keywords

Examples

			G.f.: A(x) = x - 2*x^2 - 2*x^3 + 10*x^4 + 14*x^5 - 86*x^6 - 126*x^7 + 858*x^8 + 1302*x^9 - 9378*x^10 - 14606*x^11 + ...

Crossrefs

Cf. A000081, A004111, A045648, A049075, A073075, A115593, A306768, A307365, A307366.

Programs

  • Mathematica
    terms = 28; A[] = 0; Do[A[x] = x Exp[Sum[2 A[-x^(2 k - 1)]/(2 k - 1), {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[((1 + x^k)/(1 - x^k))^((-1)^k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = 0; Table[a[n], {n, 0, 28}]

Formula

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} ((1 + x^n)/(1 - x^n))^((-1)^n*a(n)).
Recurrence: a(n+1) = (2/n) * Sum_{k=1..n} ( Sum_{d|k, k/d odd} (-1)^d*d*a(d) ) * a(n-k+1).

A308245 G.f.: x * Product_{k>=1} 1/(1 - a(k)*(-x)^k)^((-1)^k).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 34, 76, 168, 368, 838, 1964, 4544, 10464, 24658, 58984, 140072, 331456, 795834, 1932228, 4665304, 11227280, 27305882, 66953236, 163418448, 397826496, 976658846, 2412163316, 5935476672, 14576596320, 36023097266, 89458468968
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2019

Keywords

Crossrefs

Cf. A032305, A045648, A049075, A093637, A308246.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - a[k] (-x)^k)^((-1)^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 33}]
a[n_] := a[n] = Sum[Sum[(-1)^(k + d) d a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[a[n], {n, 1, 33}]

Formula

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k+d)*d*a(d)^(k/d) ) * a(n-k+1).

A308246 G.f.: x * Product_{k>=1} (1 + a(k)*(-x)^k)^((-1)^k).

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 44, 104, 246, 620, 1600, 4082, 10436, 27360, 73046, 193296, 509984, 1371214, 3727792, 10065872, 27145058, 74142688, 204005440, 558475342, 1527058912, 4213709856, 11694035010, 32331790700, 89266126856, 248240818282, 693599213260
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2019

Keywords

Crossrefs

Cf. A032305, A045648, A049075, A093637, A308245.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[x Product[(1 + a[k] (-x)^k)^((-1)^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 31}]
a[n_] := a[n] = Sum[Sum[(-1)^(k/d + k + d + 1) d a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[a[n], {n, 1, 31}]

Formula

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+k+d+1)*d*a(d)^(k/d) ) * a(n-k+1).

A345232 G.f. A(x) satisfies: A(x) = x + x^2 / exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Original entry on oeis.org

1, 1, -1, -1, 2, 0, -4, 3, 7, -12, -8, 35, -6, -87, 84, 172, -367, -187, 1175, -417, -3003, 3621, 5723, -15126, -4374, 47813, -26192, -119731, 175835, 211797, -699210, -57982, 2148031, -1601079, -5161935, 9125489, 8093890, -34478125, 3997517, 101971205, -97182026
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A007562, A045648, A345233.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, 1, -add(add(a(n-k)*
      d*a(d), d=numtheory[divisors](k)), k=1..n-2)/(n-2))
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Jun 11 2021
  • Mathematica
    nmax = 41; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[-A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; a[n_] := a[n] = -(1/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 41}]

Formula

G.f.: x + x^2 * Product_{n>=1} (1 - x^n)^a(n).
a(n+2) = -(1/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).

A345878 G.f. A(x) satisfies: A(x) = x / exp(2 * Sum_{k>=1} A(x^k) / k).

Original entry on oeis.org

1, -2, 5, -18, 70, -282, 1179, -5104, 22634, -102128, 467637, -2168208, 10157664, -48005858, 228607728, -1095885048, 5284044080, -25609804110, 124693451466, -609641464746, 2991742731876, -14731354000792, 72761153346680, -360397156557696, 1789733084330806, -8909067981051118
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

Crossrefs

Cf. A000151, A005753, A045648, A345883.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, -2*add(a(n-k)*add(
      d*a(d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..26); # Alois P. Heinz, Jun 28 2021
  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = x/Exp[2 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[n_] := a[n] = -(2/(n - 1)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 26}]

Formula

G.f.: x * Product_{n>=1} (1 - x^n)^(2*a(n)).
a(n+1) = -(2/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+1).

A345883 G.f. A(x) satisfies: A(x) = x / exp(3 * Sum_{k>=1} A(x^k) / k).

Original entry on oeis.org

1, -3, 12, -64, 372, -2268, 14394, -94296, 632328, -4317846, 29925108, -209966748, 1488507931, -10645680858, 76717312932, -556528367791, 4060765734816, -29782931545368, 219444442931836, -1623585342758532, 12057148232386980, -89842712017158526, 671521130395037280
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

Crossrefs

Cf. A006964, A045648, A052757, A345878.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, -3*add(a(n-k)*
      add(d*a(d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Jun 28 2021
  • Mathematica
    nmax = 23; A[] = 0; Do[A[x] = x/Exp[3 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[n_] := a[n] = -(3/(n - 1)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 23}]

Formula

G.f.: x * Product_{n>=1} (1 - x^n)^(3*a(n)).
a(n+1) = -(3/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+1).

A363062 G.f. A(x) satisfies: A(x) = x - x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...).

Original entry on oeis.org

1, -1, -1, 0, 1, 1, -1, -2, 0, 4, 4, -5, -13, -2, 26, 30, -29, -94, -26, 189, 246, -198, -769, -302, 1512, 2228, -1372, -6691, -3425, 12672, 21046, -9503, -60776, -38353, 109719, 205330, -61001, -567518, -427145, 967914, 2045196, -314417, -5405209, -4743873, 8625547
Offset: 1

Views

Author

Ilya Gutkovskiy, May 16 2023

Keywords

Crossrefs

Cf. A007562, A045648, A345235.

Programs

  • Mathematica
    nmax = 45; A[] = 0; Do[A[x] = x - x^2 Exp[Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[2] = -1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 45}]

Formula

G.f.: x - x^2 / Product_{n>=1} (1 - x^n)^a(n).
a(1) = 1, a(2) = -1; a(n) = (1/(n - 2)) * Sum_{k=1..n-2} ( Sum_{d|k} d * a(d) ) * a(n-k).

A363294 G.f. A(x) satisfies: A(x) = x * exp( A(x)^2/x + A(-x^2)^2/(2*x^2) + A(x^3)^2/(3*x^3) + A(-x^4)^2/(4*x^4) + ... ).

Original entry on oeis.org

1, 1, 3, 10, 37, 154, 676, 3053, 14187, 67459, 326241, 1599480, 7933272, 39736160, 200700204, 1021052197, 5227501077, 26912956631, 139244637915, 723631840568, 3775598797694, 19770494002049, 103865161431895, 547291750362216, 2891718659119578, 15317429567883000
Offset: 1

Views

Author

Ilya Gutkovskiy, May 26 2023

Keywords

Crossrefs

Cf. A005750, A005754, A045648, A363293.

Programs

  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = x Exp[Sum[A[-(-x)^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
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