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 1-10 of 15 results. Next

A007562 Number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 137, 275, 541, 1098, 2208, 4521, 9240, 19084, 39451, 82113, 171240, 358794, 753460, 1587740, 3353192, 7100909, 15067924, 32044456, 68272854, 145730675, 311575140, 667221030, 1430892924, 3072925944, 6607832422, 14226665499
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

There is no planted tree on one node by definition.
Column k=2 of A144018. - Alois P. Heinz, Oct 17 2012
It appears that a(n) is also the number of locally non-intersecting unlabeled rooted trees with n nodes, where a tree is locally non-intersecting if the branches directly under of any non-leaf node have empty intersection. - Gus Wiseman, Aug 22 2018

Examples

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 20*x^8 + 36*x^9 + ...
From _Joerg Arndt_, Jun 23 2014: (Start)
The a(8) = 20 such trees have the following level sequences:
01:  [ 0 1 2 3 4 3 2 1 ]
02:  [ 0 1 2 3 3 3 2 1 ]
03:  [ 0 1 2 3 3 2 2 1 ]
04:  [ 0 1 2 3 3 2 1 1 ]
05:  [ 0 1 2 3 2 3 2 1 ]
06:  [ 0 1 2 3 2 2 2 1 ]
07:  [ 0 1 2 3 2 2 1 1 ]
08:  [ 0 1 2 3 2 1 2 1 ]
09:  [ 0 1 2 3 2 1 1 1 ]
10:  [ 0 1 2 2 2 2 2 1 ]
11:  [ 0 1 2 2 2 2 1 1 ]
12:  [ 0 1 2 2 2 1 2 1 ]
13:  [ 0 1 2 2 2 1 1 1 ]
14:  [ 0 1 2 2 1 2 2 1 ]
15:  [ 0 1 2 2 1 2 1 1 ]
16:  [ 0 1 2 2 1 1 1 1 ]
17:  [ 0 1 2 1 2 1 2 1 ]
18:  [ 0 1 2 1 2 1 1 1 ]
19:  [ 0 1 2 1 1 1 1 1 ]
20:  [ 0 1 1 1 1 1 1 1 ]
Successive levels change by at most 1 and the last level is 1, compare to the example in A000081.
(End)
From _Gus Wiseman_, Aug 22 2018: (Start)
The a(7) = 10 locally non-intersecting trees:
  (o(o(oo)))
  (o(oo(o)))
  (o(oooo))
  (oo(o(o)))
  (oo(ooo))
  (o(o)(oo))
  (ooo(oo))
  (oo(o)(o))
  (oooo(o))
  (oooooo)
(End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000081, A000837, A004111, A007560, A144018, A289509, A316470, A316473, A316475.

Programs

  • Maple
    with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d= divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): a:= n-> `if`(n<=1, n, b(n-2)): seq(a(n), n=1..40);  # Alois P. Heinz, Sep 06 2008
  • Mathematica
    etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}] + Sum[ d*p[d], {d, Divisors[n]}])/n]; b]; b = etr[a]; a[n_] := If[n <= 1, n, b[n-2]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)
purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Intersection@@#=={}&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,10}] (* Gus Wiseman, Aug 22 2018 *)
  • PARI
    {a(n) = local(A); if( n<2, n>0, A = x / (1 - x) + O(x^n); for(k=2, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k-1)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */

Formula

Shifts left 2 places under Euler transform.
G.f.: x + x^2 / (Product_{k>0} (1 - x^k)^a(k)). - Michael Somos, Oct 06 2003
a(n) ~ c * d^n / n^(3/2), where d = 2.246066877341161662499621547921... and c = 0.68490297576105466417608032... . - Vaclav Kotesovec, Jun 23 2014
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 + ...). - Ilya Gutkovskiy, Jun 11 2021

Extensions

Better description from Christian G. Bower, May 15 1998

A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 6, 2, 2, 1, 1, 1, 12, 4, 2, 2, 1, 1, 1, 25, 6, 3, 2, 2, 1, 1, 1, 52, 10, 5, 3, 2, 2, 1, 1, 1, 113, 17, 7, 4, 3, 2, 2, 1, 1, 1, 247, 29, 10, 6, 4, 3, 2, 2, 1, 1, 1, 548, 51, 17, 8, 5, 4, 3, 2, 2, 1, 1, 1, 1226, 89, 26, 12, 7, 5, 4, 3, 2, 2, 1, 1, 1
Offset: 1

Views

Author

Alois P. Heinz, Jun 23 2018

Keywords

Examples

			Triangle T(n,k) begins:
    1;
    1,  1;
    1,  1, 1;
    2,  1, 1, 1;
    3,  2, 1, 1, 1;
    6,  2, 2, 1, 1, 1;
   12,  4, 2, 2, 1, 1, 1;
   25,  6, 3, 2, 2, 1, 1, 1;
   52, 10, 5, 3, 2, 2, 1, 1, 1;
  113, 17, 7, 4, 3, 2, 2, 1, 1, 1;

Links

Crossrefs

Columns k=1-10 give: A004111, A007560, A316075, A316076, A316077, A316078, A316079, A316080, A316081, A316082.
T(2n,n) gives A000009.
Cf. A057427, A144018, A316101.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(T(i, k), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> `if`(n
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[T[i, k], j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
T[n_, k_] := If[n < k, Sign[n], b[n - k, n - k, k]];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)

A363388 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^2 / (k*x^k) ).

Original entry on oeis.org

1, 1, 1, 2, 5, 10, 28, 70, 190, 517, 1441, 4057, 11572, 33294, 96620, 282319, 830178, 2454384, 7292106, 21759413, 65185967, 195976025, 591097127, 1788122219, 5423917828, 16493458475, 50270190728, 153544874713, 469916030995, 1440807810639, 4425266768759, 13613578089594, 41943137192265
Offset: 1

Views

Author

Ilya Gutkovskiy, May 30 2023

Keywords

Crossrefs

Cf. A005754, A007560, A363387.

Programs

  • Mathematica
    nmax = 33; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 33}]
  • PARI
    seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(-sum(k=1, m, (-1)^k*subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A345234 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, 2, 3, 5, 9, 17, 31, 58, 112, 218, 427, 844, 1683, 3381, 6824, 13842, 28226, 57796, 118762, 244874, 506515, 1050688, 2185095, 4555217, 9517423, 19926174, 41798031, 87833877, 184881588, 389765182, 822901122, 1739763655, 3682955618, 7806103024, 16564348106, 35187631009
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A007560, A007562, A049075, A345235.

Programs

  • Mathematica
    nmax = 38; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[(-1)^(k + 1) 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[(-1)^(k + 1) Sum[(-1)^(k/d + d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 38}]

Formula

G.f.: x + x^2 * Product_{n>=1} (1 + (-x)^n)^((-1)^n*a(n)).
a(n+2) = (1/n) * Sum_{k=1..n} (-1)^(k+1) * ( Sum_{d|k} (-1)^(k/d+d) * d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 2.21094707842288180828190718521597733363607957468229824761... and c = 0.664585976397397791197984310778764361056468131968... - Vaclav Kotesovec, Jun 19 2021

A345235 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, 3, 5, 8, 14, 25, 44, 78, 142, 261, 479, 886, 1655, 3105, 5843, 11043, 20965, 39938, 76285, 146123, 280691, 540475, 1042885, 2016481, 3906647, 7582034, 14739395, 28697969, 55958110, 109262713, 213619535, 418158580, 819491034, 1607764395, 3157551026, 6207346544
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A007560, A007562, A045648, A345234.

Programs

  • Mathematica
    nmax = 40; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[(-1)^(k + 1) 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[(-1)^k Sum[(-1)^d d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 40}]

Formula

G.f.: x + x^2 / Product_{n>=1} (1 - (-x)^n)^((-1)^n*a(n)).
a(n+2) = (1/n) * Sum_{k=1..n} (-1)^k * ( Sum_{d|k} (-1)^d * d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 2.04187801797233390910633071122033289228232310618876458... and c = 0.624667034123125135463988884805660643637934291759335... - Vaclav Kotesovec, Jun 19 2021

A345243 G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} (-1)^(k+1) * A(x^k) / k).

Original entry on oeis.org

1, 1, 2, 3, 8, 17, 42, 107, 272, 719, 1914, 5163, 14088, 38733, 107370, 299511, 840372, 2370020, 6714316, 19100096, 54534696, 156230943, 448942998, 1293692305, 3737568960, 10823759093, 31413810702, 91358248179, 266193726712, 776989772307, 2271695757714, 6652074198889
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A005753, A007560, A345244, A345245.

Programs

  • Mathematica
    nmax = 32; A[] = 0; Do[A[x] = x + x^2 Exp[2 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] = (2/(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, 32}]

Formula

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

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).

A345245 G.f. A(x) satisfies: A(x) = x + x^2 * exp(4 * Sum_{k>=1} (-1)^(k+1) * A(x^k) / k).

Original entry on oeis.org

1, 1, 4, 10, 36, 135, 504, 2000, 8072, 33099, 138132, 582930, 2485412, 10692219, 46340984, 202175344, 887175352, 3913032212, 17338327848, 77141235796, 344491008296, 1543591834950, 6937783312048, 31270131096820, 141305878384704, 640065923118435, 2905664234243052, 13217615913137250
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A007560, A052772, A345243, A345244.

Programs

  • Mathematica
    nmax = 28; A[] = 0; Do[A[x] = x + x^2 Exp[4 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] = (4/(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, 28}]

Formula

G.f.: x + x^2 * Product_{n>=1} (1 + x^n)^(4*a(n)).
a(n+2) = (4/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d * a(d) ) * a(n-k+2).

A363467 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^3 / (k*x^(2*k)) ).

Original entry on oeis.org

1, 1, 1, 3, 9, 25, 88, 292, 1031, 3685, 13433, 49608, 185465, 699963, 2664650, 10217130, 39428179, 153009240, 596761737, 2337875430, 9195732624, 36301739221, 143780858517, 571191310205, 2275409450019, 9087376470138, 36377539265376, 145937953205705, 586645566919856
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 03 2023

Keywords

Crossrefs

Cf. A007560, A052755, A363388, A363465, A363468.

Programs

  • Mathematica
    nmax = 29; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^3/(k x^(2 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[a[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1) d g[d + 2], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 29}]

A363468 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} (-1)^(k+1) * A(x^k)^4 / (k*x^(3*k)) ).

Original entry on oeis.org

1, 1, 1, 4, 14, 48, 201, 812, 3455, 14961, 65954, 294884, 1334526, 6098879, 28114885, 130561444, 610244889, 2868547475, 13552299256, 64316483918, 306473091394, 1465727378317, 7033293786125, 33851816310445, 163384902125185, 790589562321385, 3834540111072545, 18638976010097900
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 03 2023

Keywords

Crossrefs

Cf. A007560, A052775, A363388, A363466, A363467.

Programs

  • Mathematica
    nmax = 28; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[(-1)^(k + 1) A[x^k]^4/(k x^(3 k)), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; f[n_] := f[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; g[n_] := g[n] = Sum[f[k] f[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[(-1)^(k/d + 1)d g[d + 3], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 28}]
Showing 1-10 of 15 results. Next

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