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 1-4 of 4 results.

A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

Original entry on oeis.org

1, 2, 7, 20, 59, 162, 449, 1200, 3194, 8348, 21646, 55480, 141152, 356056, 892284, 2221208, 5497945, 13533858, 33151571, 80826748, 196219393, 474425518, 1142758067, 2742784304, 6561052331, 15645062126, 37194451937, 88174252924, 208463595471, 491585775018
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _Geoffrey Critzer_, Mar 07 2012: (Start)
Per comment in A102866, a(n) is also the number of multisets of binary words of total length n.
a(2) = 7 because the multisets are {a,a}, {b,b}, {a,b}, {aa}, {ab}, {ba}, {bb};
a(3) = 20 because the multisets are {a,a,a}, {b,b,b}, {a,a,b}, {a,b,b}, {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb};
where the words within each multiset are separated by commas. (End)

Links

Crossrefs

Cf. A034691, the Euler transform of 1, 2, 4, 8, 16, 32, 64, ...
Cf. A075729, A102866.
Column k=2 of A144074.
Row sums of A055375 and of A209406.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(2^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018 ~
  • Maple
    series(1/product((1-x^(n))^(2^(n)),n=1..20),x=0,12); (Wieder)
# second Maple program:
with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*2^d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 02 2011
  • Mathematica
    nn = 20; p = Product[1/(1 - x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(2^k))) \\ G. C. Greubel, Nov 09 2018

Formula

G.f.: 1/Product_{n>0} (1-x^n)^(2^n). - Thomas Wieder, Mar 06 2005
a(n) ~ c^2 * 2^(n-1) * exp(2*sqrt(n) - 1/2) / (sqrt(Pi) * n^(3/4)), where c = A247003 = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... . - Vaclav Kotesovec, Mar 09 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - 2*x^k))). - Ilya Gutkovskiy, Nov 09 2018

Extensions

More terms from Thomas Wieder, Mar 06 2005

A209406 Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.

Original entry on oeis.org

2, 4, 3, 8, 8, 4, 16, 26, 12, 5, 32, 64, 44, 16, 6, 64, 164, 132, 62, 20, 7, 128, 384, 376, 200, 80, 24, 8, 256, 904, 1008, 623, 268, 98, 28, 9, 512, 2048, 2632, 1792, 870, 336, 116, 32, 10, 1024, 4624, 6624, 5040, 2632, 1117, 404, 134, 36, 11
Offset: 1

Views

Author

Geoffrey Critzer, Mar 08 2012

Keywords

Comments

Equivalently, T(n,k) is the number of partitions of the integer n with two types of 1's, four types of 2's, ..., 2^i types of i's...; having exactly k summands (of any type).
Row sums = A034899.

Examples

			Triangle T(n,k) begins:
    2;
    4,    3;
    8,    8,    4;
   16,   26,   12,    5;
   32,   64,   44,   16,   6;
   64,  164,  132,   62,  20,   7;
  128,  384,  376,  200,  80,  24,   8;
  256,  904, 1008,  623, 268,  98,  28,  9;
  512, 2048, 2632, 1792, 870, 336, 116, 32, 10;
  ...

Links

Crossrefs

Cf. A034899, A055375, A208741, A290222, A292506.
T(2n,n) gives A359962.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(2^i+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017
  • Mathematica
    nn = 10; p[x_, y_] := Product[1/(1 - y x^i)^(2^i), {i, 1, nn}]; f[list_] := Select[lst, # > 0 &]; Map[f, Drop[CoefficientList[Series[p[x, y], {x, 0, nn}], {x, y}], 1]] // Flatten

Formula

O.g.f.: Product_{i>=1} 1/(1-y*x^i)^(2^i).

A360634 Number T(n,k) of sets of nonempty words over binary alphabet with a total of n letters of which k are the first letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 2, 6, 6, 2, 2, 11, 16, 11, 2, 3, 18, 37, 37, 18, 3, 4, 28, 73, 100, 73, 28, 4, 5, 42, 133, 228, 228, 133, 42, 5, 6, 61, 227, 470, 593, 470, 227, 61, 6, 8, 86, 370, 899, 1370, 1370, 899, 370, 86, 8, 10, 119, 580, 1617, 2894, 3497, 2894, 1617, 580, 119, 10
Offset: 0

Views

Author

Alois P. Heinz, Feb 14 2023

Keywords

Examples

			T(4,0) = 2: {bbbb}, {b,bbb}.
T(4,1) = 11: {abbb}, {babb}, {bbab}, {bbba}, {a,bbb}, {ab,bb}, {abb,b}, {b,bab}, {b,bba}, {ba,bb}, {a,b,bb}.
T(4,2) = 16: {aabb}, {abab}, {abba}, {baab}, {baba}, {bbaa}, {a,abb}, {a,bab}, {a,bba}, {aa,bb}, {aab,b}, {ab,ba}, {aba,b}, {b,baa}, {a,ab,b}, {a,b,ba}.
Triangle T(n,k) begins:
   1;
   1,   1;
   1,   3,   1;
   2,   6,   6,    2;
   2,  11,  16,   11,    2;
   3,  18,  37,   37,   18,    3;
   4,  28,  73,  100,   73,   28,    4;
   5,  42, 133,  228,  228,  133,   42,    5;
   6,  61, 227,  470,  593,  470,  227,   61,   6;
   8,  86, 370,  899, 1370, 1370,  899,  370,  86,   8;
  10, 119, 580, 1617, 2894, 3497, 2894, 1617, 580, 119, 10;
  ...

Links

Crossrefs

Columns k=0-2 give: A000009, A095944, A360650.
Row sums give A102866.
T(2n,n) gives A360638.
Cf. A055375 (the same for multisets), A200751, A208741.

Programs

  • Maple
    g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i), k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);
  • Mathematica
    g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)

Formula

T(n,k) = T(n,n-k).
Sum_{k=0..2n} (-1)^k*T(2n,k) = A200751(n). - Alois P. Heinz, Sep 09 2023

A360626 Number of multisets of nonempty words over binary alphabet where each letter occurs n times.

Original entry on oeis.org

1, 3, 21, 131, 830, 5066, 30456, 179256, 1038593, 5928071, 33402561, 186021335, 1025162709, 5596047683, 30282832593, 162573152651, 866385400935, 4585861723905, 24120596727003, 126124094912499, 655868112470175, 3393060517486981, 17468543071082489
Offset: 0

Views

Author

Alois P. Heinz, Feb 14 2023

Keywords

Examples

			a(0) = 1: {}.
a(1) = 3: {ab}, {ba}, {a,b}.
a(2) = 21: {aabb}, {abab}, {abba}, {baab}, {baba}, {bbaa}, {a,abb}, {a,bab}, {a,bba}, {aa,bb}, {aab,b}, {ab,ab}, {ab,ba}, {aba,b}, {b,baa}, {ba,ba}, {a,a,bb}, {a,ab,b}, {a,b,ba}, {aa,b,b}, {a,a,b,b}.

Links

Crossrefs

Cf. A055375, A359962, A360638 (the same for sets).

Programs

  • Maple
    g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..31);
  • Mathematica
    g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i] + k - 1, k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

Formula

a(n) = [x^(2n)*y^n] Product_{i>=1} Product_{j=0..i} 1/(1-x^i*y^j)^binomial(i,j).
a(n) = A055375(2n,n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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