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.

A055375 Euler transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 14, 21, 14, 5, 7, 26, 48, 48, 26, 7, 11, 45, 103, 131, 103, 45, 11, 15, 75, 198, 312, 312, 198, 75, 15, 22, 120, 366, 674, 830, 674, 366, 120, 22, 30, 187, 637, 1359, 1961, 1961, 1359, 637, 187, 30, 42, 284, 1078, 2584, 4302, 5066, 4302, 2584, 1078, 284, 42
Offset: 0

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Author

Christian G. Bower, May 16 2000

Keywords

Comments

Number of partitions of n objects, k of which are black, into parts each of which is a sequence of objects. E.g. T(3,1) = 7; the partitions are [BWW], [WBW], [WWB], [BW,W], [WB,W], [WW,B] and [B,W,W]. - Franklin T. Adams-Watters, Jan 10 2007

Examples

			Triangle begins
   1;
   1,  1;
   2,  3,   2;
   3,  7,   7,   3;
   5, 14,  21,  14,   5;
   7, 26,  48,  48,  26,   7;
  11, 45, 103, 131, 103,  45, 11;
  15, 75, 198, 312, 312, 198, 75, 15;
  ...

Links

Crossrefs

Row sums give A034899.
Columns k=0-1 give: A000041, A014153(n-1) for n>=1.
T(2n,n) gives A360626.
Cf. A007318, A034691, A209406, A360634.

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:
T:= (n, k)-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Feb 14 2023
  • Mathematica
    nmax = 10; pp = Product[Product[1/(1 - x^i*y^j)^Binomial[i, j], {j, 0, i}], {i, 1, nmax}]; t[n_, k_] := SeriesCoefficient[pp, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2017 *)

Formula

G.f.: Product_{i>=1} Product_{j=0..i} 1/(1 - x^i y^j)^C(i,j). - Franklin T. Adams-Watters, Jan 10 2007
Sum_{k=0..2n} (-1)^k * T(2n,k) = A034691(n). - Alois P. Heinz, Dec 05 2023

A359962 Number of multisets of n nonempty words with a total of 2n letters over binary alphabet.

Original entry on oeis.org

1, 4, 26, 132, 623, 2632, 10500, 39384, 141659, 490100, 1644186, 5366436, 17113433, 53454528, 163963312, 494786352, 1471423866, 4318092136, 12520027756, 35901819336, 101909674398, 286575107424, 798886300056, 2209115439664, 6062818714752, 16522049256656
Offset: 0

Views

Author

Alois P. Heinz, Jan 20 2023

Keywords

Examples

			a(0) = 1: {}.
a(1) = 4: {aa}, {ab}, {ba}, {bb}.
a(2) = 26: {a,aaa}, {a,aab}, {a,aba}, {a,abb}, {a,baa}, {a,bab}, {a,bba}, {a,bbb}, {aa,aa}, {aa,ab}, {aa,ba}, {aa,bb}, {aaa,b}, {aab,b}, {ab,ab}, {ab,ba}, {ab,bb}, {aba,b}, {abb,b}, {b,baa}, {b,bab}, {b,bba}, {b,bbb}, {ba,ba}, {ba,bb}, {bb,bb}.

Links

Crossrefs

Cf. A209406, A360626.

Formula

a(n) = [x^(2*n)*y^n] Product_{j>=1} 1/(1-y*x^j)^(2^j).
a(n) = A209406(2n,n).

A360638 Number of sets of nonempty words over binary alphabet where each letter occurs n times.

Original entry on oeis.org

1, 3, 16, 100, 593, 3497, 20316, 116378, 658214, 3679450, 20350028, 111459648, 605060633, 3257784589, 17408647968, 92378535290, 487031130699, 2552197485757, 13298890952222, 68930923717598, 355507581655752, 1824924721216084, 9326440815314046, 47464093855706540
Offset: 0

Views

Author

Alois P. Heinz, Feb 14 2023

Keywords

Examples

			a(0) = 1: {}.
a(1) = 3: {ab}, {ba}, {a,b}.
a(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}.

Links

Crossrefs

Cf. A080277, A360626 (the same for multisets), A360634.

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:
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], {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, Dec 09 2023, after Alois P. Heinz *)

Formula

a(n) = A360634(2n,n).
a(n) mod 2 = 1 <=> n in { A080277 } U {0}.
Showing 1-3 of 3 results.

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