A235999 -id:A235999 - cp's OEIS frontend

A236002 Number of overcompositions of n.

1, 2, 4, 12, 26, 60, 144, 324, 728, 1602, 3576, 7808, 17068, 36908, 79520, 170704, 364794, 777036, 1649456, 3491188, 7367544, 15513336, 32584648, 68307264, 142904080, 298448914, 622235060, 1295320004, 2692583916, 5589586996, 11588905844, 23999052692

Offset: 0

Omar E. Pol, Jan 19 2014

nonn

Analog to overpartitions, here an overcomposition is defined to be a composition in which the first occurrence of each distinct number may be overlined (see example).
Also 1 together with the row sums of A235999.
For the number of partitions of n see A000041.
For the number of compositions of n see A011782.
For the number of overpartitions of n see A015128.
Note that there are several orderings of overcompositions, the same as the orderings of compositions, but apparently for every ordering of overcompositions there are also several suborderings according to the arrangements of the overlined parts. The same for overpartitions. See one of them in Example section.

			For n = 4 the 26 overcompositions of 4 are: [4], [4'], [1,3], [1',3], [1,3'], [1',3'], [2,2], [2',2], [1,1,2], [1',1,2], [1,1,2'], [1',1,2'], [3,1], [3',1], [3,1'], [3',1'], [1,2,1], [1',2,1], [1,2',1], [1',2',1], [2,1,1], [2',1,1], [2,1',1], [2',1',1], [1,1,1,1], [1',1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A011782, A015128, A235998, A235999, A238439.

a(n) = Sum_{k=1..A003056(n)} 2^k*A235998(n,k), n >= 1.

a(7) corrected and more terms added, Joerg Arndt, Jan 20 2014

a(19)-a(31) from Alois P. Heinz, Jan 20 2014

A235790 Triangle read by rows: T(n,k) = 2^k*A116608(n,k), n>=1, k>=1.

Original entry on oeis.org

2, 4, 4, 4, 6, 8, 4, 20, 8, 24, 8, 4, 44, 16, 8, 52, 40, 6, 68, 80, 8, 88, 120, 16, 4, 108, 200, 32, 12, 116, 296, 80, 4, 148, 416, 160, 8, 176, 536, 320, 8, 176, 776, 480, 32, 10, 220, 936, 832, 64, 4, 236, 1232, 1232, 160, 12, 272, 1472, 1872, 320

Offset: 1

Omar E. Pol, Jan 18 2014

It appears that T(n,k) is the number of overpartitions of n having k distinct parts. (This is true by definition, Joerg Arndt, Jan 20 2014).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
The first element of column k is A000079(k).

			Triangle begins:
2;
4;
4,    4;
6,    8;
4,   20;
8,   24,    8;
4,   44,   16;
8,   52,   40;
6,   68,   80;
8,   88,  120,   16;
4,  108,  200,   32;
12, 116,  296,   80;
4,  148,  416,  160;
8,  176,  536,  320;
8,  176,  776,  480,   32;
10, 220,  936,  832,   64;
4,  236, 1232, 1232,  160;
12, 272, 1472, 1872,  320;
4,  284, 1880, 2592,  640;
12, 324, 2216, 3632, 1152;
8,  328, 2704, 4944, 1856, 64;
...

Alois P. Heinz, Rows n = 1..500, flattened

Row sums give A015128, n >= 1.
Column 1 is A062011.
Cf. A000217, A003056, A116608, A196020, A211971, A235792, A235793, A235797, A235798, A235999, A236000, A236001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1)+add(x*b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n->(p->seq(2^i*coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 20 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[b[n, i-1] + Sum[x*b[n-i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[p, Table[2^i * Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Oct 20 2016, after Alois P. Heinz *)

A235998 Triangle read by rows: T(n,k) is the number of compositions of n having k distinct parts (n>=1, 1<=k<=floor((sqrt(1+8*n)-1)/2)).

Original entry on oeis.org

1, 2, 2, 2, 3, 5, 2, 14, 4, 22, 6, 2, 44, 18, 4, 68, 56, 3, 107, 146, 4, 172, 312, 24, 2, 261, 677, 84, 6, 396, 1358, 288, 2, 606, 2666, 822, 4, 950, 5012, 2226, 4, 1414, 9542, 5304, 120, 5, 2238, 17531, 12514, 480, 2, 3418, 32412, 27904, 1800, 6, 5411, 58995, 61080, 5580

Offset: 1

Omar E. Pol, Jan 19 2014

Row n has length A003056(n) hence the first element of column k is in row A000217(k).
The equivalent sequence for partitions is A116608.
For the number of compositions of n see A011782.
For the connection to overcompositions see A235999.
Row sums give A011782(n), n >= 1.
First column is A000005, second column is A131661.
T(k*(k+1)/2,k) = T(A000217(k),k) = A000142(k) = k!. - Alois P. Heinz, Jan 20 2014

			Triangle begins:
  1;
  2;
  2,    2;
  3,    5;
  2,   14;
  4,   22,     6;
  2,   44,    18;
  4,   68,    56;
  3,  107,   146;
  4,  172,   312,    24;
  2,  261,   677,    84;
  6,  396,  1358,   288;
  2,  606,  2666,   822;
  4,  950,  5012,  2226;
  4, 1414,  9542,  5304,  120;
  5, 2238, 17531, 12514,  480;
  2, 3418, 32412, 27904, 1800;
  6, 5411, 58995, 61080, 5580;
  ...

Alois P. Heinz, Rows n = 1..500, flattened

Cf. A003056, A116608, A235790, A235999, A236002.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      expand(add(b(n-i*j, i-1, p+j)/j!*`if`(j=0, 1, x), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 0)):
seq(T(n), n=1..25); # Alois P. Heinz, Jan 20 2014, revised May 25 2014

b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Sum[b[n-i*j, i-1, p+ j]/j!*If[j==0, 1, x], {j, 0, n/i}]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 1, 25}] // Flatten (* Jean-François Alcover, Dec 10 2015, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jan 19 2014

A236625 Total number of parts in all overcompositions of n.

Original entry on oeis.org

0, 2, 6, 24, 66, 180, 496, 1272, 3202, 7798, 18980, 45076, 106288, 246956, 568776, 1299184, 2944654, 6630660, 14838606, 33026000, 73126376, 161198136, 353812612, 773645124, 1685548792, 3660364490, 7924414752, 17107225340, 36832846344, 79107019964, 169505684844

Offset: 0

Omar E. Pol, Feb 01 2014

nonn

For the definition of overcomposition see A236002.
The equivalent sequence for overpartitions is A235792.
Row sums of triangle A236628.

			For n = 3 the 12 overcompositions of 3 are [3], [3'], [1, 2], [1', 2], [1, 2'], [1', 2'], [2, 1], [2', 1], [2, 1'], [2', 1'], [1, 1, 1], [1', 1, 1]. There are 24 parts, so a(3) = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001792, A006128, A235999, A236002, A235792, A236626, A236628, A236633.

b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
      `if`(i<1, 0, add((p-> p+[0, p[1]*j])(1/j!*
      `if`(j>0, 2, 1)*b(n-i*j, i-1, p+j)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..35);  # Alois P. Heinz, Apr 28 2016

b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0}, If[i < 1, {0, 0}, Sum[# + {0, #[[1]]*j}&[1/j!*If[j > 0, 2, 1]*b[n - i*j, i - 1, p + j]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 03 2022, after Alois P. Heinz *)

a(6)-a(30) from Alois P. Heinz, Feb 02 2014

A236626 Sum of all parts of all overcompositions of n.

Original entry on oeis.org

2, 8, 36, 104, 300, 864, 2268, 5824, 14418, 35760, 85888, 204816, 479804, 1113280, 2560560, 5836704, 13209612, 29690208, 66332572, 147350880, 325780056, 716862256, 1571067072, 3429697920, 7461222850, 16178111560, 34973640108, 75392349648

Offset: 1

Omar E. Pol, Feb 01 2014

nonn

For the definition of overcomposition see A236002.

The equivalent sequence for overpartitions is A235793.

			For n = 3 the 12 overcompositions of 3 are [3], [3'], [1, 2], [1', 2], [1, 2'], [1', 2'], [2, 1], [2', 1], [2, 1'], [2', 1'], [1, 1, 1], [1', 1, 1], hence the sum of all parts is 3+3+1+2+1+2+1+2+1+2+2+1+2+1+2+1+2+1+1+1+1+1+1+1 = 3*12 = 36, so a(3) = 36.

Cf. A001787, A066186, A235793, A235999, A236002, A236625, A236628, A236633.

a(n) = n*A236002(n).

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A236002 Number of overcompositions of n.

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A235790 Triangle read by rows: T(n,k) = 2^k*A116608(n,k), n>=1, k>=1.

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A235998 Triangle read by rows: T(n,k) is the number of compositions of n having k distinct parts (n>=1, 1<=k<=floor((sqrt(1+8*n)-1)/2)).

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A236625 Total number of parts in all overcompositions of n.

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A236626 Sum of all parts of all overcompositions of n.

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