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.

A182984 Total number of parts that are not the smallest part in all partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 9, 19, 29, 48, 73, 114, 161, 241, 340, 479, 662, 917, 1237, 1678, 2231, 2965, 3901, 5114, 6629, 8588, 11036, 14129, 17983, 22823, 28790, 36238, 45381, 56674, 70502, 87453, 108077, 133259, 163762, 200747, 245378, 299261
Offset: 0

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Author

Omar E. Pol, Jul 15 2011

Keywords

Comments

a(n) = sum of 2nd largest part in all partitions of n (if all parts are equal, then we assume that 0 is also a part). Example: a(5) = 6 because the sum of the 2nd largest parts in the partitions [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] is 0 + 1 + 2 + 1 + 1 + 1 + 0 = 6. - Emeric Deutsch, Dec 11 2015

Examples

			a(5) = 6 because the partitions of 5 are [5], [(4),1], [(3),2], [(3),1,1], [(2),(2),1], [(2),1,1,1] and [1,1,1,1,1], containing a total of 6 parts that are not the smallest part (shown between parentheses).

Links

Crossrefs

Cf. A006128, A092269, A116686, A264402.

Programs

  • Maple
    g := sum((sum(x^(q+i)/(1-x^q), q = i+1 .. 80))/(product(1-x^q, q = i .. 80)), i = 1 .. 80): gser := series(g, x = 0,50): seq(coeff(gser, x, n), n = 0 .. 47); # Emeric Deutsch, Nov 14 2015

Formula

a(n) = A006128(n) - A092269(n), for n >= 1.
G.f.: g(x) = Sum(Sum(x^{q+i}/(1-x^q), q=i+1..infinity)/Product(1-x^q, q=i..infinity), i=1..infinity). - Emeric Deutsch, Nov 14 2015
a(n) = Sum(k*A264402(n,k), k>=1). - Emeric Deutsch, Dec 11 2015

A268189 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the parts larger than the smallest part is k (n>=1, 0<=k<=n-1).

Original entry on oeis.org

1, 2, 0, 2, 0, 1, 3, 0, 1, 1, 2, 0, 1, 2, 2, 4, 0, 1, 1, 3, 2, 2, 0, 1, 2, 3, 3, 4, 4, 0, 1, 1, 3, 3, 6, 4, 3, 0, 1, 2, 2, 4, 5, 6, 7, 4, 0, 1, 1, 4, 2, 7, 5, 10, 8, 2, 0, 1, 2, 2, 4, 5, 7, 9, 12, 12, 6, 0, 1, 1, 3, 2, 7, 5, 11, 10, 17, 14, 2, 0, 1, 2, 3, 4, 4, 8, 8, 13, 15, 20, 21
Offset: 1

Views

Author

Emeric Deutsch, Feb 01 2016

Keywords

Comments

Sum of entries in row n is A000041(n).
T(n,0) = A000005(n) = number of divisors of n.
Sum_{k>0} k*T(n,k) = A213359(n).

Examples

			T(5,3) = 2 because in the partitions [1,1,3] and [2,3] of 5 the sum of the parts larger than the smallest part is 3.
Triangle starts:
  1;
  2, 0;
  2, 0, 1;
  3, 0, 1, 1;
  2, 0, 1, 2, 2;
  4, 0, 1, 1, 3, 2;
  2, 0, 1, 2, 3, 3, 4;
  4, 0, 1, 1, 3, 3, 6, 4;
  3, 0, 1, 2, 2, 4, 5, 6,  7;
  4, 0, 1, 1, 4, 2, 7, 5, 10,  8;
  2, 0, 1, 2, 2, 4, 5, 7,  9, 12, 12;
  6, 0, 1, 1, 3, 2, 7, 5, 11, 10, 17, 14;

Links

Crossrefs

Cf. A000005, A000041, A213359, A264402.

Programs

  • Maple
    g := add(x^i/((1-x^i)*mul(1-t^j*x^j, j = i+1 .. 100)), i = 1 .. 100); gser := simplify(series(g, x = 0, 30)); for n to 18 do P[n] := sort(coeff(gser, x, n)) end do; for n to 18 do seq(coeff(P[n], t, j), j = 0 .. n-1) end do
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(irem(n, i)=0, 1, 0)
       +`if`(i>1, add(b(n-i*j, i-1)*x^(i*j), j=0..(n-1)/i), 0))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Feb 04 2016
  • Mathematica
    b[n_, i_] := b[n, i] = Expand[If[Mod[n, i] == 0, 1, 0] + If[i > 1, Sum[b[n - i*j, i - 1]*x^(i*j), {j, 0, (n - 1)/i}], 0]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n - 1}]][b[n, n]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 21 2016, after Alois P. Heinz *)

Formula

G.f.: G(t,x) = Sum_{i>0} x^i/((1-x^i)*Product_{j>i} (1-t^j*x^j)).

A265245 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the squares of the parts is k (n>=0, k>=0).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Emeric Deutsch, Dec 06 2015

Keywords

Comments

Number of entries in row n = 1 + n^2.
Sum of entries in row n = A000041(n).
Sum(k*T(n,k), k>=0) = A066183(n).

Examples

			Row 3 is 0,0,0,1,0,1,0,0,0,1 because in the partitions of 3, namely [1,1,1], [2,1], [3], the sums of the squares of the parts are 3, 5, and 9, respectively.
Triangle starts:
1;
0,1;
0,0,1,0,1;
0,0,0,1,0,1,0,0,0,1;
0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,1.

Links

Crossrefs

Cf. A000041, A066183, A229325 - A229332, A264402, A265247 - A265253.

Programs

  • Maple
    g := 1/(product(1-t^(k^2)*x^k, k = 1 .. 100)): gser := simplify(series(g, x = 0, 15)): for n from 0 to 8 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 8 do seq(coeff(P[n], t, j), j = 0 .. n^2) end do; # yields sequence in triangular form
  • Mathematica
    m = 8; CoefficientList[#, t]& /@ CoefficientList[1/Product[(1 - t^(k^2)* x^k), {k, 1, m}] + O[x]^m, x] // Flatten (* Jean-François Alcover, Feb 19 2019 *)

Formula

G.f.: G(t,x) = 1/Product_{k>=1} (1 - t^{k^2}*x^k).
Showing 1-3 of 3 results.

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