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.

A116482 Triangle read by rows: T(n,k) is the number of partitions of n having k even parts (n>=0, 0<=k<=floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 1, 4, 4, 2, 1, 5, 6, 3, 1, 6, 8, 5, 2, 1, 8, 11, 7, 3, 1, 10, 14, 10, 5, 2, 1, 12, 19, 14, 7, 3, 1, 15, 24, 19, 11, 5, 2, 1, 18, 31, 26, 15, 7, 3, 1, 22, 39, 34, 21, 11, 5, 2, 1, 27, 49, 45, 29, 15, 7, 3, 1, 32, 61, 58, 39, 22, 11, 5, 2, 1, 38, 76, 75, 52, 30
Offset: 0

Views

Author

Emeric Deutsch, Feb 17 2006

Keywords

Comments

Row n has 1 + floor(n/2) terms. Row sums are the partition numbers (A000041).
Column 0 yields A000009. Column 1 yields A038348. Column 2 yields A096778.
Sum_{k=0..floor(n/2)}k*T(n,k) = A066898(n).
From Gregory L. Simay, Nov 02 2015: (Start)
If n<=2k+1, T(n+2k,k) = A000041(n), the number of partitions of n.
T(n+2k,k) = the convolution of A000009(n-2j),which are the strict partitions of (n-2j), and p(j+k,k), which are the number of partitions of j+k having exactly k parts.
T(n+2k,k) = e(n,k) where e(n,0)= A000009(n) and e(n,k) = e(n,k-1) + e(n-2k,k-1) + e(n-4k,k-1) + ... .(End)

Examples

			T(7,2) = 3 because we have [4,2,1], [3,2,2] and [2,2,1,1,1].
Triangle starts:
   1;
   1;
   1,  1;
   2,  1;
   2,  2,  1;
   3,  3,  1;
   4,  4,  2,  1;
   5,  6,  3,  1;
   6,  8,  5,  2,  1;
   8, 11,  7,  3,  1;
  10, 14, 10,  5,  2, 1;
  12, 19, 14,  7,  3, 1;
  15, 24, 19, 11,  5, 2, 1;
  18, 31, 26, 15,  7, 3, 1;
  22, 39, 34, 21, 11, 5, 2, 1;
  27, 49, 45, 29, 15, 7, 3, 1;
Added entries for n=8 through n=15. - _Gregory L. Simay_, Nov 03 2015
From _Gregory L. Simay_, Nov 03 2015: (Start)
T(15,4) = T(7+2*4,4) = p(7) = 15, since 7 < 2*4 + 1.
T(15,3) = T(13,2) + T(9,3) = 26 + 3 = 29.
T(10,1) = T(8+2*1,1) = T(8,0) + T(6,0) + T(4,0) + T(2,0) + T(0,0) = 6 + 4 + 2 + 1 + 1 = 14.
T(15,3) = T(9+2*3) = e(9,3) = e(9,2) + e(3,2) = (e(9,1) + e(5,1) + e(1,1)) + e(3,1) = q(9) + q(7) + q(5) + q(3) + q(1) + q(5) + q(3) + q(1) + q(1) + q(3) + q(1) = q(9) + q(7) + 2*q(5) + 3*q(3) + 4*q(1) = 8 + 5 + 2*3 + 3*2 + 4*1 = 29 = the convolution sum of q(9-2j) with p(3+j,3).
(End)

Links

Crossrefs

Cf. A000041, A038348, A066898, A103919.

Programs

  • Maple
    g:=1/product((1-x^(2*j-1))*(1-t*x^(2*j)),j=1..20): gser:=simplify(series(g,x=0,22)): P[0]:=1: for n from 1 to 18 do P[n]:=coeff(gser,x^n) od: for n from 0 to 18 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; local j; if n=0 then 1 elif i<1
      then 0 else []; for j from 0 to n/i do zip((x, y)->x+y, %,
      [`if`(irem(i, 2)=0, 0$j, [][]), b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n, n):
seq (T(n), n=0..30);  # Alois P. Heinz, Jan 07 2013
  • Mathematica
    nn=8;CoefficientList[Series[Product[1/(1-x^(2i-1))/(1-y x^(2i)),{i,1,nn}],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, Jan 07 2013 *)

Formula

G.f.: G(t,x) = 1/Product_{j>=1}((1-x^(2j-1))(1-tx^(2j))).
From Gregory L. Simay, Nov 03 2015: (Start)
G.f.: T(n+2k,k) = g.f.: e(n,k) = Product_{j>=1}(1-x^2*(k+j))*p(x), where p(x) is the g.f. of the partitions of x. If n<=2k+1, then the g.f. reduces to p(x).
T(n+2k,k) = T(n+2k-2,k-1) + T(n,k).
(End)

A246579 G.f.: x^(k^2)/(mul(1-x^(2*i),i=1..k)*mul(1+x^(2*r-1),r=1..oo)) with k=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 2, -3, 5, -7, 11, -15, 21, -29, 39, -52, 69, -90, 116, -150, 190, -241, 303, -379, 470, -583, 716, -878, 1071, -1302, 1575, -1902, 2285, -2739, 3273, -3899, 4631, -5489, 6486, -7647, 8996, -10557, 12363, -14450, 16853, -19618, 22798, -26441
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

References

  • Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=3.

Crossrefs

k=0,1,2 give (apart perhaps from signs) A081360, A038348, A096778. Cf. A246590.

Programs

  • Maple
    fU:=proc(k) local a,i,r;
a:=x^(k^2)/mul(1-x^(2*i),i=1..k);
a:=a/mul(1+x^(2*r-1),r=1..101);
series(a,x,101);
seriestolist(%);
end;
fU(3);

A246580 G.f.: x^(k^2)/(mul(1-x^(2*i),i=1..k)*mul(1+x^(2*r-1),r=1..oo)) with k=4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 2, -3, 5, -7, 11, -15, 22, -30, 41, -55, 74, -97, 127, -165, 212, -271, 344, -434, 544, -680, 843, -1043, 1283, -1573, 1919, -2336, 2829, -3419, 4116, -4942, 5914, -7062, 8405, -9983, 11825, -13976, 16479, -19392, 22767
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

References

  • Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=4.

Crossrefs

k=0,1,2 give (apart perhaps from signs) A081360, A038348, A096778. Cf. A246589.

Programs

  • Maple
    fU:=proc(k) local a,i,r;
a:=x^(k^2)/mul(1-x^(2*i),i=1..k);
a:=a/mul(1+x^(2*r-1),r=1..101);
series(a,x,101);
seriestolist(%);
end;
fU(4);

A366747 Irregular triangular array, read by rows: T(n,k) = out-degree of k-th vertex in the distance graph of the strict partitions of n, where the parts of partitions and the list of partitions are in reverse-lexicographic order (Mathematica order).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 3, 1, 2, 2, 1, 1, 1, 1, 2, 3, 1, 3, 2, 1, 2, 2, 1, 1, 1, 2, 3, 1, 3, 2, 2, 3, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 3, 2, 3, 3, 2, 1, 2, 4, 1, 2, 2, 1, 1, 1, 2, 3, 1, 3, 2, 3, 3, 2
Offset: 1

Views

Author

Clark Kimberling, Oct 25 2023

Keywords

Comments

See A366156 for the distance function d and A000097 for the distance graph.
Regarding reverse lexicographic order (Mathematica order, also called canonical order; see A080577).

Examples

			Triangle begins:
  0
  0
  1
  1
  1   1
  1   1   1
  1   1   2   1
  1   1   2   2   1
  1   1   2   3   1   1   1
  1   1   2   3   1   2   2   1   1
  1   1   2   3   1   3   2   1   2   2   1
  1   1   2   3   1   3   2   2   3   2   1   3   1   1
  1   1   2   3   1   3   2   3   3   2   1   2   4   1   2   2   1
Enumerate the 6 strict partitions (= vertices) of 8 as follows:
  1: 8
  2: 7,1
  3: 6,2
  4: 5,3
  5: 5,2,1
  6: 4,3,1
Call q a neighbor of p if d(p,q)=2.
The set of neighbors for vertex k, for k = 1..6, is given by
  vertex 1: {2}  (so that vertex 1 has out-degree 1)
  vertex 2: {1,3}  (out-degree 1)
  vertex 3: {2,4,5}  (out-degree 2)
  vertex 4: {3,5,6} (out-degree 2)
  vertex 5: {3,4,6} (out degree 1)
  vertex 6: {4,5} (out degree 0),
so that row 8 is 1,1,2,2,1.
(Out-degrees of 0 are excluded except for n = 1 and n = 2.)

Crossrefs

Cf. A000009, A096778 (row sums), A366597.

Programs

  • Mathematica
    c[n_] := PartitionsQ[n]; q[n_, k_] := q[n, k] =
Select[IntegerPartitions[n], DeleteDuplicates[#] == # &][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &];
t = Table[s[n, k], {n, 1, 12}, {k, 1, c[n]}];
s1[n_, k_] := Length[Select[s[n, k], # > k &]];
t1 = Join[{0, 0}, Table[s1[n, k], {n, 1, 26}, {k, 1, c[n] - 1}]];
TableForm[t1] (* array *)
Flatten[t1]   (* sequence *)
Showing 1-4 of 4 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.