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 1-7 of 7 results.

A255517 Number A(n,k) of rooted identity trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 5, 2, 0, 0, 1, 4, 12, 18, 3, 0, 0, 1, 5, 22, 64, 66, 6, 0, 0, 1, 6, 35, 156, 363, 266, 12, 0, 0, 1, 7, 51, 310, 1193, 2214, 1111, 25, 0, 0, 1, 8, 70, 542, 2980, 9748, 14043, 4792, 52, 0, 0, 1, 9, 92, 868, 6273, 30526, 82916, 91857, 21124, 113, 0
Offset: 0

Views

Author

Alois P. Heinz, Feb 24 2015

Keywords

Comments

From Vaclav Kotesovec, Feb 24 2015: (Start)
k Limit n->infinity A(n,k)^(1/n)
1 2.517540352632003890795354598463447277335981266803... = A246169
2 5.249032491228170579164952216184309265343086337648... = A246312
3 7.969494030514425004826375511986491746399264355846...
4 10.688492754969652458452048798468242930479212456958...
5 13.407087472537747579787047072702638639945914705837...
6 16.125529360448558670505097146631763969697822205298...
7 18.843901825822305757579605844910623225182677164912...
8 21.562238702430237066018783115405680041128676137631...
9 24.280555694806692616578932533497629224907619468796...
10 26.998860838916733933849490675388336975888308433826...
100 271.64425688361559470587959030374804709717287744789...
Conjecture: For big k the limit asymptotically approaches k*exp(1).
(End)

Examples

			A(3,2) = 5:
  o    o    o    o      o
  |    |    |    |     / \
  1    1    2    2    1   2
  |    |    |    |
  1    2    1    2
Square array A(n,k) begins:
  0,  0,   0,    0,    0,     0,     0, ...
  1,  1,   1,    1,    1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6, ...
  0,  1,   5,   12,   22,    35,    51, ...
  0,  2,  18,   64,  156,   310,   542, ...
  0,  3,  66,  363, 1193,  2980,  6273, ...
  0,  6, 266, 2214, 9748, 30526, 77262, ...

Links

Crossrefs

Columns k=1-10 give: A004111, A005753, A052757, A052772, A052797, A255518, A255519, A255520, A255521, A255522.
Rows n=0-4 give: A000004, A000012, A001477, A000326, 2*A051662(k-1) for k>0.
Lower diagonal gives A255523.
Cf. A242249, A256068.

Programs

  • Maple
    with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, add(A(n-j, k)*add(
      k*A(d, k)*d*(-1)^(j/d+1), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
  • Mathematica
    A[n_, k_] := A[n, k] = If[n<2, n, Sum[A[n-j, k]*Sum[k*A[d, k]*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A038080 Number of identity trees with 3-colored nodes.

Original entry on oeis.org

1, 3, 3, 9, 39, 189, 981, 5490, 31674, 189954, 1170126, 7382745, 47494197, 310712808, 2061987642, 13855192866, 94113385437, 645424668666, 4464027720900, 31110200069511, 218292811705458, 1541172223659249
Offset: 0

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A000220, A038077-A038079, A052757, A255517.

Formula

G.f.: B(x) - B^2(x)/2 - B(x^2)/2, where B(x) is g.f. for A038079.
a(n) ~ c * d^n / n^(5/2), where d = 7.969494030514425004826375511986491746399264355846412073489715938... and c = 0.3712461766927875417276388215355520756010680416348018056669... - Vaclav Kotesovec, Dec 26 2020

A038079 Number of rooted identity trees with 3-colored nodes.

Original entry on oeis.org

3, 9, 36, 192, 1089, 6642, 42129, 275571, 1844028, 12567762, 86924745, 608612814, 4305284400, 30723593751, 220917003516, 1599013642359, 11641144003245, 85186969539435, 626245146267393, 4622806865390823, 34251800667936939, 254640579219212457
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Links

Crossrefs

Cf. A004111, A038077-A038080.
Cf. A052757.

Formula

Shifts left and divides by 3 under Weigh transform.
a(n) = 3 * A052757(n).

A345244 G.f. A(x) satisfies: A(x) = x + x^2 * exp(3 * Sum_{k>=1} (-1)^(k+1) * A(x^k) / k).

Original entry on oeis.org

1, 1, 3, 6, 19, 57, 177, 586, 1950, 6642, 22990, 80400, 284346, 1014237, 3644841, 13185810, 47976382, 175458798, 644630064, 2378084209, 8805524949, 32714828733, 121917589291, 455625246297, 1707142362234, 6411576477380, 24133229559243, 91023263056629, 343964618949140, 1302098673500514
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A007560, A052757, A345243, A345245.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = x + x^2 Exp[3 Sum[(-1)^(k + 1) 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] = (3/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]

Formula

G.f.: x + x^2 * Product_{n>=1} (1 + x^n)^(3*a(n)).
a(n+2) = (3/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k+2).

A363475 G.f. satisfies A(x) = exp( 3 * Sum_{k>=1} (-1)^(k+1) * A(-x^k) * x^k/k ).

Original entry on oeis.org

1, 3, -6, -44, 96, 918, -2073, -22278, 52629, 597627, -1451736, -17065641, 42205373, 508415817, -1273766637, -15623442097, 39528583206, 491601500847, -1253383246330, -15759867676416, 40430096479776, 512914242127868, -1322511998532891
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2023

Keywords

Links

Crossrefs

Cf. A306768, A363474.
Cf. A052757, A363471.

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(3*sum(k=1, i, (-1)^(k+1)*subst(A, x, -x^k)*x^k/k)+x*O(x^n))); Vec(A);

Formula

A(x) = Sum_{k>=0} a(k) * x^k = Product_{k>=0} (1+x^(k+1))^(3 * (-1)^k * a(k)).
a(0) = 1; a(n) = (3/n) * Sum_{k=1..n} ( Sum_{d|k} d * (-1)^(d+k/d) * a(d-1) ) * a(n-k).

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).

A345885 G.f. A(x) satisfies: A(x) = x * exp(3 * Sum_{k>=1} (-1)^k * A(x^k) / k).

Original entry on oeis.org

1, -3, 15, -82, 486, -3090, 20497, -140010, 979131, -6976603, 50461716, -369533691, 2734423934, -20414010219, 153571115619, -1163003999342, 8859172575069, -67835214598017, 521824159637718, -4030828937892966, 31252886542570119, -243142210911325273, 1897466281615297698
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

Crossrefs

Cf. A006964, A049075, A052757, A345884.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, 3*add(a(n-k)*add(d*a(d)
      *(-1)^(k/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[(-1)^k 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[(-1)^(k/d) 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} (-1)^(k/d) * d * a(d) ) * a(n-k+1).
Showing 1-7 of 7 results.

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