A114220 -id:A114220 - cp's OEIS frontend

A357136 Triangle read by rows where T(n,k) is the number of integer compositions of n with alternating sum k = 0..n. Part of the full triangle A097805.

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 3, 0, 3, 0, 1, 0, 6, 0, 4, 0, 1, 10, 0, 10, 0, 5, 0, 1, 0, 20, 0, 15, 0, 6, 0, 1, 35, 0, 35, 0, 21, 0, 7, 0, 1, 0, 70, 0, 56, 0, 28, 0, 8, 0, 1, 126, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 0, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1

Offset: 0

Gus Wiseman, Sep 30 2022

A composition of n is a finite sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			Triangle begins:
    1
    0   1
    1   0   1
    0   2   0   1
    3   0   3   0   1
    0   6   0   4   0   1
   10   0  10   0   5   0   1
    0  20   0  15   0   6   0   1
   35   0  35   0  21   0   7   0   1
    0  70   0  56   0  28   0   8   0   1
  126   0 126   0  84   0  36   0   9   0   1
    0 252   0 210   0 120   0  45   0  10   0   1
  462   0 462   0 330   0 165   0  55   0  11   0   1
    0 924   0 792   0 495   0 220   0  66   0  12   0   1
For example, row n = 5 counts the following compositions:
  .  (32)     .  (41)   .  (5)
     (122)       (113)
     (221)       (212)
     (1121)      (311)
     (2111)
     (11111)

The full triangle counting compositions by alternating sum is A097805.
The version for partitions is A103919, full triangle A344651.
This is the right-half of even-indexed rows of A260492.
The triangle without top row and left column is A108044.
Ranking and counting compositions:
- product = sum: A335404, counted by A335405.
- sum = twice alternating sum: A348614, counted by A262977.
- length = alternating sum: A357184, counted by A357182.
- length = absolute value of alternating sum: A357185, counted by A357183.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
Cf. A000120, A051159, A070939, A114220, A114901, A242882, A262046.

Prepend[Table[If[EvenQ[nn],Prepend[#,0],#]&[Riffle[Table[Binomial[nn,k],{k,Floor[nn/2],nn}],0]],{nn,0,10}],{1}]

A357182 Number of integer compositions of n with the same length as their alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 1, 3, 1, 4, 6, 20, 13, 48, 50, 175, 141, 512, 481, 1719, 1491, 5400, 4929, 17776, 15840, 57420, 52079, 188656, 169989, 617176, 559834, 2033175, 1842041, 6697744, 6085950, 22139780, 20123989, 73262232, 66697354, 242931321, 221314299, 806516560

Offset: 0

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(1) = 1 through a(8) = 6 compositions:
  (1)  (31)  (113)  (42)  (124)  (53)
             (212)        (223)  (1151)
             (311)        (322)  (2141)
                          (421)  (3131)
                                 (4121)
                                 (5111)

For product instead of length we have A114220.
For sum equal to twice alternating sum we have A262977, ranked by A348614.
For product equal to sum we have A335405, ranked by A335404.
For absolute value we have A357183.
These compositions are ranked by A357184.
The case of partitions is A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A261983 counts non-anti-run compositions.
A357136 counts compositions by alternating sum.
Cf. A000120, A032020, A070939, A106356, A114901, A131044, A178470, A233564, A242882, A262046, A301987.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==ats[#]&]],{n,0,15}]

a(21)-a(39) from Alois P. Heinz, Sep 29 2022

A357189 Number of integer partitions of n with the same length as alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 2, 2, 4, 3, 5, 6, 9, 9, 13, 16, 23, 23, 34, 37, 54, 54, 78, 84, 120, 121, 170, 182, 252, 260, 358, 379, 517, 535, 725, 764, 1030, 1064, 1427, 1494, 1992, 2059, 2733, 2848, 3759, 3887, 5106, 5311, 6946, 7177, 9345, 9701, 12577, 12996, 16788

Offset: 0

Gus Wiseman, Sep 30 2022

nonn

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(4) = 1 through a(13) = 9 partitions:
  31   311   42   322   53     333     64     443     75       553
                  421   5111   432     5221   542     5331     652
                               531     6211   641     6222     751
                               51111          52211   6321     52222
                                              62111   7311     53311
                                                      711111   62221
                                                               63211
                                                               73111
                                                               7111111

For product equal to sum we have A001055, compositions A335405.
For product instead of length we have A004526, compositions A114220.
The version for compositions is A357182, ranked by A357184.
For sum equal to twice alternating sum we have A357189 (this sequence).
These partitions are ranked by A357486.
The reverse version is A357487, ranked by A357485.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
Cf. A051159, A070939, A131044, A262046, A262977, A301987, A357183.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[#]&]],{n,0,30}]

A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

Original entry on oeis.org

0, 1, 9, 19, 22, 28, 34, 69, 74, 84, 104, 132, 135, 141, 153, 177, 225, 265, 271, 274, 283, 286, 292, 307, 310, 316, 328, 355, 358, 364, 376, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 553, 562, 593, 610, 673, 706, 833, 898, 1041, 1047, 1053, 1058

Offset: 1

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)
  153: (3,1,3,1)
  177: (2,1,4,1)
  225: (1,1,5,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For product equal to sum we have A335404, counted by A335405.
For sum equal to twice alternating sum we have A348614, counted by A262977.
These compositions are counted by A357182.
For absolute value we have A357184, counted by A357183.
The case of partitions is counted by A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==ats[stc[#]]&]

A357488 Number of integer partitions of 2n - 1 with the same length as alternating sum.

Original entry on oeis.org

1, 0, 1, 2, 4, 5, 9, 13, 23, 34, 54, 78, 120, 170, 252, 358, 517, 725, 1030, 1427, 1992, 2733, 3759, 5106, 6946, 9345, 12577, 16788, 22384, 29641, 39199, 51529, 67626, 88307, 115083, 149332, 193383, 249456, 321134, 411998, 527472, 673233, 857539, 1089223, 1380772

Offset: 1

Gus Wiseman, Oct 02 2022

nonn

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(1) = 1 through a(7) = 9 partitions:
  (1)  .  (311)  (322)  (333)    (443)    (553)
                 (421)  (432)    (542)    (652)
                        (531)    (641)    (751)
                        (51111)  (52211)  (52222)
                                 (62111)  (53311)
                                          (62221)
                                          (63211)
                                          (73111)
                                          (7111111)

For product equal to sum we have A001055, compositions A335405.
The version for compositions appears to be A222763, odd version of A357182.
These are the odd-indexed terms of A357189, ranked by A357486.
These partitions are ranked by the odd-sum portion of A357485.
Except at the start, alternately adding zeros gives A357487.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
Cf. A004526, A051159, A114220, A131044, A262046, A262977, A357183, A357184.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[#]&]],{n,1,30,2}]

a(n) = A357189(2n - 1).

More terms from Alois P. Heinz, Oct 04 2022

A309049 Number T(n,k) of (binary) max-heaps on n elements from the set {0,1} containing exactly k 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 4, 2, 1, 1, 1, 4, 6, 6, 5, 2, 1, 1, 1, 4, 7, 8, 7, 5, 2, 1, 1, 1, 5, 10, 12, 11, 8, 5, 2, 1, 1, 1, 5, 11, 16, 17, 13, 9, 5, 2, 1, 1, 1, 6, 15, 23, 27, 24, 16, 10, 5, 2, 1, 1, 1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1

Offset: 0

Alois P. Heinz, Jul 09 2019

Also the number T(n,k) of (binary) min-heaps on n elements from the set {0,1} containing exactly k 1's.

The sequence of column k satisfies a linear recurrence with constant coefficients of order A063915(k).

			T(6,0) = 1: 111111.
T(6,1) = 3: 111011, 111101, 111110.
T(6,2) = 4: 110110, 111001, 111010, 111100.
T(6,3) = 4: 101001, 110010, 110100, 111000.
T(6,4) = 2: 101000, 110000.
T(6,5) = 1: 100000.
T(6,6) = 1: 000000.
T(7,4) = T(7,7-3) = A000108(3) = 5: 1010001, 1010010, 1100100, 1101000, 1110000.
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  1,  1;
  1, 2,  2,  1,  1;
  1, 3,  3,  2,  1,  1;
  1, 3,  4,  4,  2,  1,  1;
  1, 4,  6,  6,  5,  2,  1,  1;
  1, 4,  7,  8,  7,  5,  2,  1,  1;
  1, 5, 10, 12, 11,  8,  5,  2,  1, 1;
  1, 5, 11, 16, 17, 13,  9,  5,  2, 1, 1;
  1, 6, 15, 23, 27, 24, 16, 10,  5, 2, 1, 1;
  1, 6, 16, 27, 34, 34, 27, 18, 11, 5, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap

Columns k=0-10 give: A000012, A110654, A114220 (for n>0), A326504, A326505, A326506, A326507, A326508, A326509, A326510, A326511.
Row sums give A091980(n+1).
T(2n,n) gives A309050.
Rows reversed converge to A000108.
Cf. A000225, A000295, A003095, A024358, A056971, A063915, A137560, A168542, A202019, A309051, A309052.

b:= proc(n) option remember; `if`(n=0, 1, (g-> (f-> expand(
      x^n+b(f)*b(n-1-f)))(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);

b[n_] := b[n] = If[n == 0, 1, Function[g, Function[f, Expand[x^n + b[f]*b[n - 1 - f]]][Min[g - 1, n - g/2]]][2^Floor[Log[2, n]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Oct 04 2019, after Alois P. Heinz *)

Sum_{k=0..n}    k  * T(n,k) = A309051(n).
Sum_{k=0..n} (n-k) * T(n,k) = A309052(n).
Sum_{k=0..2^n-1} T(2^n-1,k) = A003095(n+1).
Sum_{k=0..2^n-1} (2^n-1-k) * T(2^n-1,k) = A024358(n).
Sum_{k=0..n} (T(n,k) - T(n-1,k)) = A168542(n).
T(m,m-n) = A000108(n) for m >= 2^n-1 = A000225(n).
T(2^n-1,k) = A202019(n+1,k+1).

A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 2, 3, 2, 5, 12, 22, 26, 58, 100, 203, 282, 616, 962, 2045, 2982, 6518, 9858, 21416, 31680, 69623, 104158, 228930, 339978, 751430, 1119668, 2478787, 3684082, 8182469, 12171900, 27082870, 40247978, 89748642, 133394708, 297933185, 442628598, 990210110

Offset: 0

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The a(1) = 1 through a(8) = 12 compositions:
  (1)  (13)  (113)  (24)  (124)  (35)
       (31)  (212)  (42)  (151)  (53)
             (311)        (223)  (1115)
                          (322)  (1151)
                          (421)  (1214)
                                 (1313)
                                 (1412)
                                 (1511)
                                 (2141)
                                 (3131)
                                 (4121)
                                 (5111)

For product instead of length we have A114220.
For sum equal to twice alternating sum we have A262977, ranked by A348614.
For product equal to sum we have A335405, ranked by A335404.
This is the absolute value version of A357182.
These compositions are ranked by A357185.
The case of partitions is A357189.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A261983 counts non-anti-run compositions.
A357136 counts compositions by alternating sum.
Cf. A000120, A032020, A070939, A106356, A114901, A131044, A178470, A233564, A242882, A262046, A301987.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==Abs[ats[#]]&]],{n,0,15}]

a(21)-a(39) from Alois P. Heinz, Sep 29 2022

A055802 a(n) = T(n,n-2), array T as in A055801.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21, 22, 28, 29, 36, 37, 45, 46, 55, 56, 66, 67, 78, 79, 91, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435

Offset: 2

Clark Kimberling, May 28 2000

For n>2, a(n)+a(n+1) seems to be A002620(n+1)+1.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A002620, A134519.

Cf. A055801, A055803, A055804, A055805, A055806.

Concatenation([1], List([3..65], n-> (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 )); # G. C. Greubel, Jan 23 2020

[1] cat [(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16: n in [3..65]]; // G. C. Greubel, Jan 23 2020

seq( `if`(n==2, 1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16), n=2..65); # G. C. Greubel, Jan 23 2020

CoefficientList[Series[(1 -2*x^2 +x^3 +2*x^4 -x^5)/((1-x)^3*(1+x)^2), {x,0,65}], x] (* Wesley Ivan Hurt, Jan 20 2017 *)
Table[If[n==2,1, (2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16], {n,2,65}] (* G. C. Greubel, Jan 23 2020 *)

Vec(x^2*(1-2*x^2+x^3+2*x^4-x^5)/((1-x)^3*(1+x)^2) + O(x^65)) \\ Charles R Greathouse IV, Feb 03 2013

vector(65, n, my(m=n+1); if(m==2, 1, (2*m^2 -6*m +11 +(-1)^m*(2*m -11))/16)) \\ G. C. Greubel, Jan 23 2020

[1]+[(2*n^2 -6*n +11 +(-1)^n*(2*n -11))/16 for n in (3..65)] # G. C. Greubel, Jan 23 2020

G.f.: x^2*(1 -2*x^2 +x^3 +2*x^4 -x^5)/((1-x)^3*(1+x)^2).
a(n) = A114220(n-1), n>=3. - R. J. Mathar, Feb 03 2013
From Colin Barker, Jan 27 2016: (Start)
a(n) = (2*n^2 +2*(-1)^n*n -6*n -11*(-1)^n +11)/16 for n>2.
a(n) = (n^2 - 2*n)/8 for n>2 and even.
a(n) = (n^2 - 4*n + 11)/8 for n odd. (End)
E.g.f.: (4*x*(x-2) + x*(x-3)*cosh(x) + (x^2 -x +11)*sinh(x))/8. - G. C. Greubel, Jan 23 2020

A357487 Number of integer partitions of n with the same length as reverse-alternating sum.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 2, 0, 4, 0, 5, 0, 9, 0, 13, 0, 23, 0, 34, 0, 54, 0, 78, 0, 120, 0, 170, 0, 252, 0, 358, 0, 517, 0, 725, 0, 1030, 0, 1427, 0, 1992, 0, 2733, 0, 3759, 0, 5106, 0, 6946, 0, 9345, 0, 12577, 0, 16788, 0, 22384, 0, 29641, 0

Offset: 0

Gus Wiseman, Oct 01 2022

nonn

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.

			The a(1) = 1 through a(13) = 9 partitions:
  1   .  .  .  311   .  322   .  333     .  443     .  553
                        421      432        542        652
                                 531        641        751
                                 51111      52211      52222
                                            62111      53311
                                                       62221
                                                       63211
                                                       73111
                                                       7111111

For product equal to sum we have A001055, compositions A335405.
The version for compositions is A357182, reverse ranked by A357184.
The reverse version is A357189, ranked by A357486.
These partitions are ranked by A357485.
Removing zeros gives A357488.
A000041 counts partitions, strict A000009.
A025047 counts alternating compositions.
A103919 counts partitions by alternating sum, full triangle A344651.
A357136 counts compositions by alternating sum, full triangle A097805.
Cf. A004526, A051159, A114220, A131044, A262046, A262977, A301987, A357183.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[Reverse[#]]&]],{n,0,30}]

A357185 Numbers k such that the k-th composition in standard order has the same length as the absolute value of its alternating sum.

Original entry on oeis.org

0, 1, 9, 12, 19, 22, 28, 34, 40, 69, 74, 84, 97, 104, 132, 135, 141, 144, 153, 177, 195, 198, 204, 216, 225, 240, 265, 271, 274, 283, 286, 292, 307, 310, 316, 321, 328, 355, 358, 364, 376, 386, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 544, 553

Offset: 1

Gus Wiseman, Sep 28 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   12: (1,3)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   40: (2,4)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
   97: (1,5,1)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For sum equal to twice alternating sum we have A348614, counted by A262977.
For product equal to sum we have A335404, counted by A335405.
These compositions are counted by A357183.
This is the absolute value version of A357184, counted by A357183.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
A357136 counts compositions by alternating sum.
Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==Abs[ats[stc[#]]]&]

cp's OEIS Frontend

A357136 Triangle read by rows where T(n,k) is the number of integer compositions of n with alternating sum k = 0..n. Part of the full triangle A097805.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A357182 Number of integer compositions of n with the same length as their alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A357189 Number of integer partitions of n with the same length as alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A357488 Number of integer partitions of 2n - 1 with the same length as alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A309049 Number T(n,k) of (binary) max-heaps on n elements from the set {0,1} containing exactly k 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A357183 Number of integer compositions with the same length as the absolute value of their alternating sum.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A055802 a(n) = T(n,n-2), array T as in A055801.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs