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 1-5 of 5 results.

A276434 Sum over all partitions of n of the number of distinct parts i of multiplicity i+1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 5, 6, 10, 12, 19, 23, 34, 41, 58, 72, 98, 121, 162, 200, 262, 323, 415, 511, 650, 796, 1000, 1222, 1522, 1851, 2287, 2771, 3399, 4103, 5000, 6015, 7289, 8735, 10530, 12579, 15094, 17968, 21468, 25477, 30319, 35873, 42531, 50177, 59291
Offset: 0

Views

Author

Emeric Deutsch, Sep 30 2016

Keywords

Examples

			a(6) = 3 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2], [1',1,2,2], [2',2,2], [1,1,1,3], [1,2,3], [3,3], [1',1,4], [2,4], [1,5], [6] of 6 only the marked parts satisfy the requirement.

Links

Crossrefs

Cf. A276427, A276428, A276429, A276433, A277099, A277100, A277101, A277102.

Programs

  • Maple
    g := (sum(x^(i*(i+1))*(1-x^i), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i+1<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 30 2016
  • Mathematica
    max = 60; s = Sum[x^(i*(i+1))*(1-x^i), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 08 2016 *)

Formula

a(n) = Sum(k*A276433(n,k), k>=0).
G.f.: g(x) = Sum_(i>=1) (x^(i(i+1))(1-x^i))/Product_(i>=1) (1-x^i).

A277101 Sum over all partitions of n of the number of distinct parts i of multiplicity i - 1.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 4, 5, 8, 10, 15, 20, 29, 37, 52, 67, 89, 115, 152, 192, 251, 316, 405, 508, 644, 799, 1006, 1243, 1546, 1901, 2351, 2871, 3527, 4289, 5232, 6336, 7688, 9264, 11189, 13430, 16137, 19299, 23097, 27514, 32799, 38944, 46246, 54738, 64782, 76430, 90171
Offset: 0

Views

Author

Emeric Deutsch, Oct 10 2016

Keywords

Examples

			a(6) = 4 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2'], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2',3], [3',3], [1,1,4], [2',4], [1,5], [6] of 6 only the marked parts satisfy the requirement.

Links

Crossrefs

Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277102.

Programs

  • Maple
    g := (sum(x^(i*(i+1))*(1-x^(i+1)), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i-1<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 10 2016
  • Mathematica
    max = 60; s = Sum[x^(i*(i+1))*(1-x^(1+i)), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 08 2016 *)

Formula

a(n) = Sum(k*A277100(n,k), k>=0).
G.f.: g(x) = Sum_(i>=1)(x^(i(i+1))(1-x^(i+1)))/Product_(i>=1)(1-x^i).

A276433 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i of multiplicity i+1 (n>=0).

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 1, 6, 1, 8, 3, 12, 3, 18, 3, 1, 24, 6, 32, 10, 45, 10, 1, 59, 17, 1, 79, 21, 1, 104, 28, 3, 137, 37, 2, 177, 50, 4, 229, 64, 4, 295, 82, 8, 377, 105, 8, 477, 139, 10, 1, 605, 174, 13, 761, 220, 21, 956, 275, 24, 1193, 350, 31, 1
Offset: 0

Views

Author

Emeric Deutsch, Sep 30 2016

Keywords

Comments

Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A277099(n).
Sum(k*T(n,k), k>=0) = A276434(n).

Examples

			The partition [1,1,3,3,3,3,4] has 2 parts i of multiplicity i+1: 1 and 3.
T(5,1) = 1, counting [1,1,3].
T(6,1) = 3, counting [1,1,4], [1,1,2,2], and [2,2,2].
T(8,2) = 1, counting [1,1,2,2,2].
Triangle starts:
1;
1;
1,1;
3;
4,1;
6,1;
8,3.

Links

Crossrefs

Cf. A000041, A276427, A276428, A276429, A276434, A277099, A277100, A277101, A277102.

Programs

  • Maple
    G := mul((t-1)*x^(i*(i+1))+1/(1-x^i), i = 1 .. 100): Gser := simplify(series(G, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n],t,k),k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i+1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Sep 30 2016
  • Mathematica
    b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[If[i + 1 == j, x, 1]*b[n - i*j, i - 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Nov 28 2016 after Alois P. Heinz *)

Formula

G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^i)).

A277099 Number of partitions of n containing no part i of multiplicity i+1.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 8, 12, 18, 24, 32, 45, 59, 79, 104, 137, 177, 229, 295, 377, 477, 605, 761, 956, 1193, 1484, 1840, 2276, 2800, 3441, 4210, 5141, 6261, 7603, 9206, 11132, 13419, 16144, 19380, 23223, 27763, 33134, 39467, 46931, 55703, 66008, 78085, 92239, 108776, 128091, 150617
Offset: 0

Views

Author

Emeric Deutsch, Sep 30 2016

Keywords

Examples

			a(4) = 4 because we have [1,1,1,1], [1,3], [2,2], and [4]; the partition [1,1,2] does not qualify.

Links

Crossrefs

Cf. A276427, A276428, A276429, A276433, A276434, A277100, A277101, A277102.

Programs

  • Maple
    g:= product(1/(1-x^i)-x^(i*(i+1)), i = 1 .. 100): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i+1=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 30 2016
  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1/(1-x^k) - x^(k*(k+1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 30 2016 *)

Formula

a(n) = A276433(n,0).
G.f.: g(x) = Product_{i>=1} (1/(1-x^i) - x^(i*(i+1))).

A277100 Irregular triangle read by rows: T(n,k) is the number of partitions of n having k distinct parts i (i>=2) of multiplicity i-1 (n>=0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 5, 2, 7, 4, 10, 5, 15, 6, 1, 21, 8, 1, 28, 13, 1, 37, 18, 1, 50, 25, 2, 67, 31, 3, 88, 42, 5, 115, 55, 6, 150, 73, 8, 193, 93, 11, 248, 122, 15, 317, 154, 19, 402, 200, 24, 1, 508, 253, 30, 1, 640, 320, 41, 1, 802, 399, 53, 1, 1002, 503, 69, 1
Offset: 0

Views

Author

Emeric Deutsch, Oct 10 2016

Keywords

Comments

Sum of entries in row n is A000041(n) (the partition numbers).
T(n,0) = A277102(n).
Sum(k*T(n,k), k>=0) = A277101(n).

Examples

			The partition [1,1,2,3,3,3,3,4,4,4] has 2 parts i of multiplicity i-1: 2 and 4.
T(5,1) = 2, counting [1,1,1,2] and [2,3].
T(8,2) = 1, counting [2,3,3].
Triangle starts:
1;
1;
1, 1;
2, 1;
4, 1;
5, 2;
7, 4;
...

Links

Crossrefs

Cf. A000041, A116608, A276427, A276428, A276429, A276433, A276434, A277099, A277101, A277102.

Programs

  • Maple
    g := mul((t-1)*x^(i*(i+1))+1/(1-x^(i+1)), i = 1 .. 100)/(1-x): gser := simplify(series(g, x = 0, 35)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 30 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(i-1=j, x, 1)*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Oct 10 2016
  • Mathematica
    b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, x, 1]*b[n-i*j, i-1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 08 2016 after Alois P. Heinz *)

Formula

G.f.: G(t,x) = Product_{i>=1} ((t-1)*x^(i(i+1)) + 1/(1-x^(i+1))).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.