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 13 results. Next

A004111 Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6299, 14426, 33209, 76851, 178618, 416848, 976296, 2294224, 5407384, 12780394, 30283120, 71924647, 171196956, 408310668, 975662480, 2335443077, 5599508648, 13446130438, 32334837886, 77863375126, 187737500013, 453203435319, 1095295264857, 2649957419351
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The nodes are unlabeled.
There is a natural correspondence between rooted identity trees and finitary sets (sets whose transitive closure is finite); each node represents a set, with the children of that node representing the members of that set. When the set corresponding to an identity tree is written out using braces, there is one set of braces for each node of the tree; thus a(n) is also the number of sets that can be made using n pairs of braces. - Franklin T. Adams-Watters, Oct 25 2011
Shifts left under WEIGH transform. - Franklin T. Adams-Watters, Jan 17 2007
Is this the sequence mentioned in the middle of page 355 of Motzkin (1948)? - N. J. A. Sloane, Jul 04 2015. Answer from David Broadhurst, Apr 06 2022: The answer is No. Motzkin was considering a sequence asymptotic to Catalan(n)/(4*n), namely A006082, which begins 1, 1, 1, 2, 3, 6, 12, 27, ... but he miscalculated and got 1, 1, 1, 2, 3, 6, 12, 25, ... instead! - N. J. A. Sloane, Apr 06 2022

Examples

			The 2 identity trees with 4 nodes are:
     O    O
    / \   |
   O   O  O
       |  |
       O  O
          |
          O
These correspond to the sets {{},{{}}} and {{{{}}}}.
G.f.: x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 12*x^7 + 25*x^8 + 52*x^9 + ...

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 330.
  • S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 301 and 562.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 64, Eq. (3.3.15); p. 80, Problem 3.10.
  • D. E. Knuth, Fundamental Algorithms, 3rd Ed., 1997, pp. 386-388.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000009, A000081, A000220, A196118, A196154, A196161, A227819.
Cf. A027750, A035056, A246169.
Column k=1 of A255517, A316074, A316101, A318757.

Programs

  • Haskell
    import Data.List (genericIndex)
a004111 = genericIndex a004111_list
a004111_list = 0 : 1 : f 1 [1] where
   f x zs = y : f (x + 1) (y : zs) where
            y = (sum $ zipWith (*) zs $ map g [1..]) `div` x
   g k = sum $ zipWith (*) (map (((-1) ^) . (+ 1)) $ reverse divs)
                           (zipWith (*) divs $ map a004111 divs)
                           where divs = a027750_row k
-- Reinhard Zumkeller, Apr 29 2014
  • Maple
    A004111 := proc(n)
        spec := [ A, {A=Prod(Z,PowerSet(A))} ]:
        combstruct[count](spec, size=n) ;
end proc:
# second Maple program:
with(numtheory):
a:= proc(n) a(n):= `if`(n<2, n, add(a(n-k)*add(a(d)*d*
       (-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 15 2014
  • Mathematica
    s[ n_, k_ ] := s[ n, k ]=a[ n+1-k ]+If[ n<2k, 0, -s[ n-k, k ] ]; a[ 1 ]=1; a[ n_ ] := a[ n ]=Sum[ a[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ a[ i ], {i, 1, 30} ] (* Robert A. Russell *)
a[ n_] := If[ n < 2, Boole[n == 1], Nest[ CoefficientList[ Normal[ Times @@ (Table[1 + x^k, {k, Length@#}]^#) + x O[x]^Length@#], x] &, {}, n - 1][[n]]]; (* Michael Somos, Jul 10 2014 *)
a[n_] := a[n] = Sum[a[n-k]*Sum[a[d]*d*(-1)^(k/d+1),{d, Divisors[k]}], {k, 1, n-1}]/(n-1); a[0]=0; a[1]=1; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 02 2015 *)
  • PARI
    N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1) * d * A[d]) * A[n-k+1] ) );
concat([0], A)
\\ Joerg Arndt, Jul 10 2014

Formula

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) d*a(d) ) * a(n-k+1). - Mitchell Harris, Dec 02 2004
G.f. satisfies A(x) = x*exp(A(x) - A(x^2)/2 + A(x^3)/3 - A(x^4)/4 + ...). [Harary and Prins]
Also A(x) = Sum_{n >= 1} a(n)*x^n = x * Product_{n >= 1} (1+x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = A246169 = 2.51754035263200389079535..., c = 0.3625364233974198712298411097408713812865256408189512533230825639621448038... . - Vaclav Kotesovec, Aug 22 2014, updated Dec 26 2020

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 *)

A144042 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Euler transform applied k times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 9, 1, 1, 5, 13, 25, 20, 1, 1, 6, 19, 51, 77, 48, 1, 1, 7, 26, 89, 197, 258, 115, 1, 1, 8, 34, 141, 410, 828, 871, 286, 1, 1, 9, 43, 209, 751, 2052, 3526, 3049, 719, 1, 1, 10, 53, 295, 1260, 4337, 10440, 15538, 10834, 1842, 1, 1, 11, 64
Offset: 1

Views

Author

Alois P. Heinz, Sep 08 2008

Keywords

Examples

			Square array begins:
    1,   1,    1,     1,     1,     1,      1,      1, ...
    1,   1,    1,     1,     1,     1,      1,      1, ...
    2,   3,    4,     5,     6,     7,      8,      9, ...
    4,   8,   13,    19,    26,    34,     43,     53, ...
    9,  25,   51,    89,   141,   209,    295,    401, ...
   20,  77,  197,   410,   751,  1260,   1982,   2967, ...
   48, 258,  828,  2052,  4337,  8219,  14379,  23659, ...
  115, 871, 3526, 10440, 25512, 54677, 106464, 192615, ...

Links

Crossrefs

Columns k=1-10 give: A000081, A007563, A144035, A144036, A144037, A144038, A144039, A144040, A144041, A305727.
Rows n=2-4 give: A000012, A000027, A034856.
Main diagonal gives A305725.
Cf. A316101.

Programs

  • Maple
    etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
        add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
      end end:
g:= proc(k) option remember; local b, t; b[0]:= j->
      `if`(j<2, j, b[k](j-1)); for t to k do
       b[t]:= etr(b[t-1]) od: eval(b[0])
    end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # revised Alois P. Heinz, Aug 27 2018
  • Mathematica
    etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table[A[n, 1 + d-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A007561 Number of asymmetric rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

0, 1, 1, 1, 3, 6, 16, 43, 120, 339, 985, 2892, 8606, 25850, 78347, 239161, 734922, 2271085, 7054235, 22010418, 68958139, 216842102, 684164551, 2165240365, 6871792256, 21865189969, 69737972975, 222915760126, 714001019626, 2291298553660, 7366035776888
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A007563, A035079-A035081.
Column k=2 of A316101.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*g(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 0, b(n-1, n-1)):
seq(a(n), n=0..40); # Alois P. Heinz, May 19 2013
  • Mathematica
    g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<1, 0, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 0, 40}] (* Jean-François Alcover, Feb 11 2014, after Alois P. Heinz *)

Formula

Shifts left when weigh-transform applied twice.
a(n) ~ c * d^n / n^(3/2), where d = 3.382016466020272807429818743..., c = 0.161800727760188847021075748... . - Vaclav Kotesovec, Jul 26 2014

Extensions

Additional comments from Christian G. Bower

A316102 a(n) is the n-th term of the sequence that shifts left by one position when Weigh transform is applied n times; a(0) = 0.

Original entry on oeis.org

0, 1, 1, 1, 5, 21, 126, 918, 7767, 73972, 784751, 9144147, 115883755, 1584170940, 23204993605, 362165818664, 5993702566315, 104749261905895, 1926269870412241, 37155958512895667, 749685441940827386, 15783243142166081008, 345955075391798325101
Offset: 0

Views

Author

Alois P. Heinz, Jun 24 2018

Keywords

Links

Crossrefs

Main diagonal of A316101.
Cf. A305725.

Formula

a(n) = A316101(n,n).

A316103 Sequence shifts left when Weigh transform is applied three times with a(n) = n for n<2.

Original entry on oeis.org

0, 1, 1, 1, 4, 10, 32, 105, 356, 1227, 4327, 15444, 55830, 203785, 750388, 2783465, 10392923, 39027561, 147308309, 558550110, 2126574259, 8126608112, 31160297271, 119847314043, 462250371050, 1787514078393, 6928792593819, 26916827827548, 104780189202901
Offset: 0

Views

Author

Alois P. Heinz, Jun 24 2018

Keywords

Links

Crossrefs

Column k=3 of A316101.
Cf. A144035.

A316104 Sequence shifts left when Weigh transform is applied four times with a(n) = n for n<2.

Original entry on oeis.org

0, 1, 1, 1, 5, 15, 55, 210, 826, 3306, 13511, 55922, 234344, 991701, 4233301, 18204973, 78803016, 343074124, 1501253960, 6599380039, 29129716885, 129056936567, 573707565933, 2558203198595, 11439372705501, 51285319846823, 230473042074581, 1038019965961144
Offset: 0

Views

Author

Alois P. Heinz, Jun 24 2018

Keywords

Links

Crossrefs

Column k=4 of A316101.
Cf. A144036.

A316105 Sequence shifts left when Weigh transform is applied five times with a(n) = n for n<2.

Original entry on oeis.org

0, 1, 1, 1, 6, 21, 86, 371, 1647, 7438, 34266, 159927, 755533, 3604684, 17347231, 84103279, 410421861, 2014380304, 9937355972, 49247272198, 245062080990, 1223998706732, 6134070857861, 30835493057217, 155444304401138, 785636444084278, 3980190985058791
Offset: 0

Views

Author

Alois P. Heinz, Jun 24 2018

Keywords

Links

Crossrefs

Column k=5 of A316101.
Cf. A144037.

A316106 Sequence shifts left when Weigh transform is applied six times with a(n) = n for n<2.

Original entry on oeis.org

0, 1, 1, 1, 7, 28, 126, 602, 2961, 14805, 75468, 389817, 2037772, 10758252, 57287568, 307324993, 1659446224, 9011958240, 49191458767, 269736352723, 1485146234680, 8207439244571, 45510032845462, 253127421288198, 1411859674966108, 7895241738604045
Offset: 0

Views

Author

Alois P. Heinz, Jun 24 2018

Keywords

Links

Crossrefs

Column k=6 of A316101.
Cf. A144038.

A316107 Sequence shifts left when Weigh transform is applied seven times with a(n) = n for n<2.

Original entry on oeis.org

0, 1, 1, 1, 8, 36, 176, 918, 4936, 26958, 150026, 846177, 4829398, 27836499, 161828375, 947782262, 5587055678, 33124063134, 197385032889, 1181572853600, 7102047195736, 42846230488269, 259358408077863, 1574775193889333, 9588601977357132, 58534511044031645
Offset: 0

Views

Author

Alois P. Heinz, Jun 24 2018

Keywords

Links

Crossrefs

Column k=7 of A316101.
Cf. A144039.
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