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 1-6 of 6 results.

A124419 Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 25, 75, 225, 780, 2704, 10556, 41209, 178031, 769129, 3630780, 17139600, 87548580, 447195609, 2452523325, 13450200625, 78697155750, 460457244900, 2859220516290, 17754399678409, 116482516809889, 764214897046969, 5277304280371714
Offset: 0

Views

Author

Emeric Deutsch, Oct 31 2006

Keywords

Examples

			a(4) = 4 because we have 13|24, 1|24|3, 13|2|4 and 1|2|3|4.

Links

Crossrefs

Cf. A000110, A088911, A124418, A124420, A124421, A124422, A124423, A152875, A274310.
Column k=0 of A124418 and of A363493.
Column k=2 of A275069.

Programs

  • Maple
    Q[0]:=1: for n from 1 to 30 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 30 do Q[n]:=Q[n] od: seq(subs({t=1,s=1,x=0},Q[n]),n=0..30);
# second Maple program:
with(combinat):
a:= n-> bell(floor(n/2))*bell(ceil(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013
  • Mathematica
    a[n_] := BellB[Floor[n/2]]*BellB[Ceiling[n/2]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2015, after Alois P. Heinz *)

Formula

a(n) = Q[n](1,1,0), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
a(n) = A000110(floor(n/2)) * A000110(ceiling(n/2)). - Alois P. Heinz, Oct 23 2013
a(n) mod 2 = A088911(n). - Alois P. Heinz, Jun 06 2023

A363519 Number T(n,k) of partitions of [n] having exactly k parity changes within the partition, n>=0, 0<=k<=max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 4, 0, 3, 4, 8, 0, 2, 18, 14, 18, 0, 7, 27, 87, 42, 40, 0, 5, 102, 162, 360, 147, 101, 0, 20, 179, 866, 931, 1456, 434, 254, 0, 15, 675, 1746, 5836, 4755, 5778, 1619, 723, 0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064, 0, 52, 5216, 19863, 93452, 117172, 206570, 115178, 94210, 20271, 6586
Offset: 0

Views

Author

Alois P. Heinz, Jun 07 2023

Keywords

Comments

The blocks are ordered with increasing least elements.

Examples

			T(4,1) = 3: 134|2, 13|24, 13|2|4.
T(4,2) = 4: 124|3, 14|23, 14|2|3, 1|24|3.
T(4,3) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
T(5,2) = 18: 1245|3, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5.
T(5,4) = 18: 12345, 1234|5, 123|45, 123|4|5, 12|345, 12|34|5, 12|3|45, 12|3|4|5, 145|23, 1|2345, 1|234|5, 1|23|45, 1|23|4|5, 145|2|3, 1|2|345, 1|2|34|5, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  1;
  0,  2;
  0,  1,    4;
  0,  3,    4,    8;
  0,  2,   18,   14,    18;
  0,  7,   27,   87,    42,    40;
  0,  5,  102,  162,   360,   147,   101;
  0, 20,  179,  866,   931,  1456,   434,   254;
  0, 15,  675, 1746,  5836,  4755,  5778,  1619,  723;
  0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064;
  ...

Links

Crossrefs

Column k=1 gives A363550.
Row sums give A000110.
T(n,max(0,n-1)) gives A274547.
Cf. A363493, A363549.

Programs

  • Maple
    b:= proc(l, i, t) option remember; expand(`if`(l=[], 1,
      add((f-> b(subsop(j=[][], l), j, `if`(f, 1-t, t))*
      `if`(f, x, 1))(l[j]=t), j=[1, $i..nops(l)])))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
         b([ seq(irem(i, 2), i=2..n)], 1, 0)):
seq(T(n), n=0..12);
  • Mathematica
    b[l_, i_, t_] := b[l, i, t] = Expand[If[l == {}, 1, Sum[Function[f, b[ReplacePart[l, j -> Nothing], j, If[f, 1 - t, t]]*If[f, x, 1]][l[[j]] == t], {j, Join[{1}, Range[i, Length@l]]}]]];
T[n_] := CoefficientList[b[ Table[Mod[i, 2], {i, 2, n}], 1, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

Formula

Sum_{k=0..max(0,n-1)} k * T(n,k) = A363549(n).

A363496 Total number of parity changes within the blocks of all partitions of [n].

Original entry on oeis.org

0, 0, 1, 4, 17, 74, 356, 1808, 9923, 57442, 354407, 2296028, 15704028, 112266048, 841442105, 6564854864, 53413489773, 450789496454, 3950844987040, 35809477617544, 335901221506491, 3250110998386534, 32453151223493139, 333520967584364248, 3528754456836294712
Offset: 0

Views

Author

Alois P. Heinz, Jun 05 2023

Keywords

Examples

			a(4) = 17 = 6*1 + 4*2 + 1*3: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34, 123|4, 12|34, 14|23, 1|234, 1234.

Links

Crossrefs

Cf. A000110, A077613, A363493, A363549.

Programs

  • Maple
    b:= proc(n, x, y) option remember; `if`(n=0, [1, 0],
     `if`(y=0, 0, (p-> p+[0, p[1]])(b(n-1, y-1, x+1)*y))+
        b(n-1, y, x)*x + b(n-1, y, x+1))
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..24);

Formula

a(n) = Sum_{k=0..max(0,n-1)} k * A363493(n,k).

A363495 Number of partitions of [2n+1] having exactly n parity changes within their blocks.

Original entry on oeis.org

1, 2, 17, 202, 3899, 98282, 3270604, 134513166, 6744026175, 400657370384, 27819913699591, 2222485356153758, 202085549223540498, 20700107045049813072, 2369116259054858660518, 300712325745715659503258, 42064844140178917094949029, 6448050588990736076081469470
Offset: 0

Views

Author

Alois P. Heinz, Jun 05 2023

Keywords

Examples

			a(0) = 1: 1.
a(1) = 2: 12|3, 1|23.
a(2) = 17: 1235|4, 123|4|5, 1245|3, 12|34|5, 125|3|4, 12|3|45, 1345|2, 134|25, 14|235, 14|23|5, 15|234, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.

Links

Crossrefs

Cf. A363493.

Programs

  • Maple
    b:= proc(n, x, y) option remember; `if`(n=0, 1,
     `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
        b(n-1, y, x)*x + b(n-1, y, x+1))
    end:
a:= n-> coeff(b(2*n+1, 0$2),z,n):
seq(a(n), n=0..17);

Formula

a(n) = A363493(2n+1,n).

A363511 Number of partitions of [n] having exactly one parity change within their blocks.

Original entry on oeis.org

0, 0, 1, 2, 6, 18, 61, 210, 778, 3008, 12219, 52268, 231726, 1083012, 5202199, 26307710, 135972580, 738339310, 4081523615, 23649300862, 139096468520, 855529383396, 5329630673249, 34643027568520, 227682351175868, 1558106351450416, 10766192988109009
Offset: 0

Views

Author

Alois P. Heinz, Jun 06 2023

Keywords

Examples

			a(4) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.

Links

Crossrefs

Column k=1 of A363493.
Cf. A000110, A124419.

Programs

  • Maple
    b:= proc(n, x, y) option remember; convert(series(
     `if`(n=0, 1, `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))
       +b(n-1, y, x)*x + b(n-1, y, x+1)), z, 2), polynom)
    end:
a:= n-> coeff(b(n, 0$2), z, 1):
seq(a(n), n=0..27);

A363588 Number of partitions of [n] having exactly two parity changes within their blocks.

Original entry on oeis.org

0, 0, 0, 1, 4, 17, 68, 292, 1252, 5670, 26114, 126073, 621914, 3206277, 16888898, 92771126, 519907322, 3032369590, 18012896770, 111155162265, 697399200274, 4537750415991, 29972920817228, 204993708306706, 1421278374189924, 10188372221843166, 73948157842293620
Offset: 0

Views

Author

Alois P. Heinz, Jun 10 2023

Keywords

Examples

			a(4) = 4: 123|4, 12|34, 14|23, 1|234.

Links

Crossrefs

Column k=2 of A363493.
Cf. A000110.

Programs

  • Maple
    b:= proc(n, x, y) option remember; convert(series(
     `if`(n=0, 1, `if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))
       +b(n-1, y, x)*x + b(n-1, y, x+1)), z, 3), polynom)
    end:
a:= n-> coeff(b(n, 0$2), z, 2):
seq(a(n), n=0..27);
Showing 1-6 of 6 results.

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