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 1-3 of 3 results.

A144074 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 20, 5, 0, 1, 5, 26, 64, 59, 7, 0, 1, 6, 40, 148, 276, 162, 11, 0, 1, 7, 57, 285, 843, 1137, 449, 15, 0, 1, 8, 77, 488, 2020, 4632, 4648, 1200, 22, 0, 1, 9, 100, 770, 4140, 13876, 25124, 18585, 3194, 30, 0, 1, 10, 126
Offset: 0

Views

Author

Alois P. Heinz, Sep 09 2008

Keywords

Comments

Column k > 1 is asymptotic to k^n * exp(2*sqrt(n) - 1/2 + c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} 1/(m*(k^(m-1)-1)). - Vaclav Kotesovec, Mar 14 2015

Examples

			A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.
A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.
A(2,3) = 15: {aa}, {a,a}, {bb}, {b,b}, {cc}, {c,c}, {ab}, {ba}, {a,b}, {ac}, {ca}, {a,c}, {bc}, {cb}, {b,c}.
A(3,2) = 20: {aaa}, {a,aa}, {a,a,a}, {bbb}, {b,bb}, {b,b,b}, {aab}, {aba}, {baa}, {a,ab}, {a,ba}, {aa,b}, {a,a,b}, {bba}, {bab}, {abb}, {b,ba}, {b,ab}, {bb,a}, {b,b,a}.
Square array begins:
  1, 1,   1,    1,    1,     1, ...
  0, 1,   2,    3,    4,     5, ...
  0, 2,   7,   15,   26,    40, ...
  0, 3,  20,   64,  148,   285, ...
  0, 5,  59,  276,  843,  2020, ...
  0, 7, 162, 1137, 4632, 13876, ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000041, A034899, A144067, A144068, A144069, A144070, A144071, A144072, A144073, A292837.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A252654.
Cf. A004248, A257740, A256142, A292804.

Programs

  • Maple
    with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k^j)(n); seq(seq(A(n, d-n), n=0..d), d=0..14);
  • Mathematica
    a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, ] = 1; a[?Positive, 0] = 0;
Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)
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}]/n]; b];
A[n_, k_] := etr[k^#&][n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

Formula

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).
Column k is Euler transform of the powers of k.
T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - Alois P. Heinz, May 08 2015

Extensions

Name changed by Alois P. Heinz, Sep 21 2018

A297569 Number of nonisomorphic proper colorings of partition star graph using five colors.

Original entry on oeis.org

5, 20, 50, 80, 100, 320, 320, 175, 800, 680, 1280, 1280, 280, 1600, 2720, 3200, 5120, 5120, 5120, 420, 2800, 6800, 4080, 6400, 20480, 10400, 12800, 20480, 20480, 20480, 600, 4480, 13600, 16320, 11200, 51200, 43520, 41600, 25600, 81920, 81920, 51200, 81920, 81920, 81920, 825, 6720, 23800, 40800, 19380, 17920, 102400, 174080, 104000, 166400, 44800, 204800, 174080, 327680, 164480, 102400, 327680, 327680, 204800, 327680, 327680
Offset: 0

Views

Author

Marko Riedel, Dec 31 2017

Keywords

Comments

A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.

Examples

			Rows are:
    5;
   20,
   50,   80;
  100,  320,  320;
  175,  800,  680, 1280, 1280;
  280, 1600, 2720, 3200, 5120, 5120, 5120;

Links

Crossrefs

Cf. A297567, A297568, A297570.
Row sums give 5*A144068.
Row lengths give A000041.

Programs

  • Maple
    b:= (n, i)-> `if`(n=0, [5], `if`(i<1, [], [seq(map(x-> x*
     binomial(4^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 14 2018
  • Mathematica
    b[n_, i_] := If[n == 0, {5}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 4^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] := b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

Formula

For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=5.

A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

Original entry on oeis.org

1, 1, 5, 21, 95, 415, 1851, 8155, 36030, 158510, 696502, 3052966, 13359230, 58346206, 254405630, 1107479694, 4813850699, 20894227355, 90567536543, 392066476815, 1695180397145, 7320927664713, 31581573600685, 136094434672509, 585876330191950, 2519701493092958
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

Crossrefs

Cf. A034691, A104460, A144068, A343350, A343351, A343352, A343353, A343354, A343355.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*4^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 12 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

a(n) ~ exp(sqrt(n) - 1/8 + c/4) * 2^(2*n - 3/2) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (4^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021
Showing 1-3 of 3 results.

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