A239830 -id:A239830 - cp's OEIS frontend

A058622 a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n/2).

0, 1, 1, 4, 5, 16, 22, 64, 93, 256, 386, 1024, 1586, 4096, 6476, 16384, 26333, 65536, 106762, 262144, 431910, 1048576, 1744436, 4194304, 7036530, 16777216, 28354132, 67108864, 114159428, 268435456, 459312152, 1073741824, 1846943453

Offset: 0

Yong Kong (ykong(AT)curagen.com), Dec 29 2000

a(n) is the number of n-digit binary sequences that have more 1's than 0's. - Geoffrey Critzer, Jul 16 2009
Maps to the number of walks that end above 0 on the number line with steps being 1 or -1. - Benjamin Phillabaum, Mar 06 2011
Chris Godsil observes that a(n) is the independence number of the (n+1)-folded cube graph; proof is by a Cvetkovic's eigenvalue bound to establish an upper bound and a direct construction of the independent set by looking at vertices at an odd (resp., even) distance from a fixed vertex when n is odd (resp., even). - Stan Wagon, Jan 29 2013
Also the number of subsets of {1,2,...,n} that contain more odd than even numbers.  For example, for n=4, a(4)=5 and the 5 subsets are {1}, {3}, {1,3}, {1,2,3}, {1,3,4}. See A014495 when same number of even and odd numbers. - Enrique Navarrete, Feb 10 2018
Also half the number of length-n binary sequences with a different number of zeros than ones. This is also the number of integer compositions of n with nonzero alternating sum, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Also the number of integer compositions of n+1 with alternating sum <= 0, ranked by A345915 (reverse: A345916). - Gus Wiseman, Jul 19 2021

			G.f. = x + x^2 + 4*x^3 + 5*x^4 + 16*x^5 + 22*x^6 + 64*x^7 + 93*x^8 + ...
From _Gus Wiseman_, Jul 19 2021: (Start)
The a(1) = 1 through a(5) = 16 compositions with nonzero alternating sum:
  (1)  (2)  (3)      (4)      (5)
            (1,2)    (1,3)    (1,4)
            (2,1)    (3,1)    (2,3)
            (1,1,1)  (1,1,2)  (3,2)
                     (2,1,1)  (4,1)
                              (1,1,3)
                              (1,2,2)
                              (1,3,1)
                              (2,1,2)
                              (2,2,1)
                              (3,1,1)
                              (1,1,1,2)
                              (1,1,2,1)
                              (1,2,1,1)
                              (2,1,1,1)
                              (1,1,1,1,1)
(End)

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.1.7)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Folded Cube Graph
Eric Weisstein's World of Mathematics, Independence Number

The odd bisection is A000302.
The even bisection is A000346.
The following relate to compositions with nonzero alternating sum:
- The complement is counted by A001700 or A138364.
- The version for alternating sum > 0 is A027306.
- The unordered version is A086543 (even bisection: A182616).
- The version for alternating sum < 0 is A294175.
- These compositions are ranked by A345921.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A345197 counts compositions by length and alternating sum.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A007318, A008549, A034871, A114121, A120452, A163493, A210736, A239830, A344611.

[(2^n -(1+(-1)^n)*Binomial(n, Floor(n/2))/2)/2: n in [0..40]]; // G. C. Greubel, Aug 08 2022

Table[Sum[Binomial[n, Floor[n/2 + i]], {i, 1, n}], {n, 0, 32}] (* Geoffrey Critzer, Jul 16 2009 *)
a[n_] := If[n < 0, 0, (2^n - Boole[EvenQ @ n] Binomial[n, Quotient[n, 2]])/2]; (* Michael Somos, Nov 22 2014 *)
a[n_] := If[n < 0, 0, n! SeriesCoefficient[(Exp[2 x] - BesselI[0, 2 x])/2, {x, 0, n}]]; (* Michael Somos, Nov 22 2014 *)
Table[2^(n - 1) - (1 + (-1)^n) Binomial[n, n/2]/4, {n, 0, 40}] (* Eric W. Weisstein, Mar 21 2018 *)
CoefficientList[Series[2 x/((1-2x)(1 + 2x + Sqrt[(1+2x)(1-2x)])), {x, 0, 40}], x] (* Eric W. Weisstein, Mar 21 2018 *)
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],ats[#]!=0&]],{n,0,15}] (* Gus Wiseman, Jul 19 2021 *)

a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n\2); \\ Michel Marcus, Dec 30 2015

my(x='x+O('x^100)); concat(0, Vec(2*x/((1-2*x)*(1+2*x+((1+2*x)*(1-2*x))^(1/2))))) \\ Altug Alkan, Dec 30 2015

from math import comb
def A058622(n): return (1<>1)>>1) if n else 0 # Chai Wah Wu, Aug 25 2025

[(2^n - binomial(n, n//2)*((n+1)%2))/2 for n in (0..40)] # G. C. Greubel, Aug 08 2022

a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n/2).
a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n, i).
G.f.: 2*x/((1-2*x)*(1+2*x+((1+2*x)*(1-2*x))^(1/2))). - Vladeta Jovovic, Apr 27 2003
E.g.f: (e^(2x)-I_0(2x))/2 where I_n is the Modified Bessel Function. - Benjamin Phillabaum, Mar 06 2011
Logarithmic derivative of the g.f. of A210736 is a(n+1). - Michael Somos, Nov 22 2014
Even index: a(2n) = 2^(n-1) - A088218(n). Odd index: a(2n+1) = 2^(2n). - Gus Wiseman, Jul 19 2021
D-finite with recurrence n*a(n) +2*(-n+1)*a(n-1) +4*(-n+1)*a(n-2) +8*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 23 2021
a(n) = 2^n-A027306(n). - R. J. Mathar, Sep 23 2021
A027306(n) - a(n) = A126869(n). - R. J. Mathar, Sep 23 2021

A344604 Number of alternating compositions of n, including twins (x,x).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 13, 19, 30, 48, 76, 118, 187, 293, 461, 725, 1140, 1789, 2815, 4422, 6950, 10924, 17169, 26979, 42405, 66644, 104738, 164610, 258708, 406588, 639010, 1004287, 1578364, 2480606, 3898600, 6127152, 9629624, 15134213, 23785389, 37381849, 58750469

Offset: 0

Gus Wiseman, May 27 2021

nonn

We define a composition to be alternating including twins (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. Except in the case of twins (x,x), all such compositions are anti-runs (A003242). These compositions avoid the weak consecutive patterns (1,2,3) and (3,2,1), the strict version being A344614.

The version without twins (x,x) is A025047 (alternating compositions).

			The a(1) = 1 through a(7) = 19 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)
       (11)  (12)  (13)   (14)   (15)    (16)
             (21)  (22)   (23)   (24)    (25)
                   (31)   (32)   (33)    (34)
                   (121)  (41)   (42)    (43)
                          (131)  (51)    (52)
                          (212)  (132)   (61)
                                 (141)   (142)
                                 (213)   (151)
                                 (231)   (214)
                                 (312)   (232)
                                 (1212)  (241)
                                 (2121)  (313)
                                         (412)
                                         (1213)
                                         (1312)
                                         (2131)
                                         (3121)
                                         (12121)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
A344605 counts alternating patterns including twins.
A344606 counts alternating permutations of prime factors including twins.
Counting compositions by patterns:
- A011782 no conditions.
- A003242 avoiding (1,1) adjacent.
- A102726 avoiding (1,2,3).
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3) adjacent.
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A238279, A239830, A333213, A238279/A333755, A344612, A344616, A344617, A344618.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,15}]

a(n > 0) = A025047(n) + 1 if n is even, otherwise A025047(n). - Gus Wiseman, Nov 03 2021

a(21)-a(40) from Alois P. Heinz, Nov 04 2021

A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 3, 1, 1, 7, 9, 6, 3, 1, 1, 11, 14, 12, 6, 3, 1, 1, 15, 23, 20, 12, 6, 3, 1, 1, 22, 34, 35, 21, 12, 6, 3, 1, 1, 30, 52, 56, 38, 21, 12, 6, 3, 1, 1, 42, 75, 91, 62, 38, 21, 12, 6, 3, 1, 1, 56, 109, 140, 103, 63, 38, 21, 12, 6, 3, 1, 1

Offset: 0

Gus Wiseman, May 31 2021

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.

Also the number of reversed integer partitions of 2n with alternating sum 2k.

			Triangle begins:
   1
   1   1
   2   1   1
   3   3   1   1
   5   5   3   1   1
   7   9   6   3   1   1
  11  14  12   6   3   1   1
  15  23  20  12   6   3   1   1
  22  34  35  21  12   6   3   1   1
  30  52  56  38  21  12   6   3   1   1
  42  75  91  62  38  21  12   6   3   1   1
  56 109 140 103  63  38  21  12   6   3   1   1
  77 153 215 163 106  63  38  21  12   6   3   1   1
Row n = 5 counts the following partitions:
  (55)          (442)        (433)      (622)    (811)  (10)
  (3322)        (541)        (532)      (721)
  (4411)        (22222)      (631)      (61111)
  (222211)      (32221)      (42211)
  (331111)      (33211)      (52111)
  (22111111)    (43111)      (4111111)
  (1111111111)  (2221111)
                (3211111)
                (211111111)

The columns with initial 0's removed appear to converge to A006330.
The odd version is A239829.
The non-reversed version is A239830.
Row sums are A344611, odd bisection of A344607.
Including odd n and negative k gives A344612 (strict: A344739).
The strict case is A344649 (row sums: A344650).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A001250, A003242, A027187, A028260, A124754, A152146, A344608, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],k==sats[#]&]],{n,0,15,2},{k,0,n,2}]

A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

Original entry on oeis.org

0, 3, 10, 13, 15, 36, 41, 43, 46, 50, 53, 55, 58, 61, 63, 136, 145, 147, 150, 156, 162, 165, 167, 170, 173, 175, 180, 185, 187, 190, 196, 201, 203, 206, 210, 213, 215, 218, 221, 223, 228, 233, 235, 238, 242, 245, 247, 250, 253, 255, 528, 545, 547, 550, 556, 568

Offset: 1

Gus Wiseman, Jun 03 2021

nonn

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

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 sequence of terms together with the corresponding compositions begins:
    0: ()
    3: (1,1)
   10: (2,2)
   13: (1,2,1)
   15: (1,1,1,1)
   36: (3,3)
   41: (2,3,1)
   43: (2,2,1,1)
   46: (2,1,1,2)
   50: (1,3,2)
   53: (1,2,2,1)
   55: (1,2,1,1,1)
   58: (1,1,2,2)
   61: (1,1,1,2,1)
   63: (1,1,1,1,1,1)
  136: (4,4)
  145: (3,4,1)
  147: (3,3,1,1)
  150: (3,2,1,2)
  156: (3,1,1,3)

The version for Heinz numbers of partitions is A000290, counted by A000041.
These are the positions of zeros in A344618.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >= 0.
A124754 gives the alternating sum of standard compositions.
A316524 is the alternating sum of the prime indices of n.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
All of the following pertain to compositions in standard order:
- The length is A000120.
- Converting to reversed ranking gives A059893.
- The rows are A066099.
- The sum is A070939.
- The runs are counted by A124767.
- The reversed version is A228351.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- The Heinz number is A333219.
- Anti-run compositions are ranked by A333489.
Cf. A000070, A000097, A003242, A006330, A028260, A119899, A239830, A344605, A344607, A344650, A344739.

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

A345197 Concatenation of square matrices A(n), each read by rows, where A(n)(k,i) is the number of compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 3, 4, 0, 0, 3, 4, 3, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 03 2021

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

			The matrices for n = 1..7:
  1   0 1   0 0 1   0 0 0 1   0 0 0 0 1   0 0 0 0 0 1   0 0 0 0 0 0 1
      1 0   1 1 0   1 1 1 0   1 1 1 1 0   1 1 1 1 1 0   1 1 1 1 1 1 0
            0 1 0   0 1 2 0   0 1 2 3 0   0 1 2 3 4 0   0 1 2 3 4 5 0
                    0 1 0 0   0 2 2 0 0   0 3 4 3 0 0   0 4 6 6 4 0 0
                              0 0 1 0 0   0 0 2 3 0 0   0 0 3 6 6 0 0
                                          0 0 1 0 0 0   0 0 3 3 0 0 0
                                                        0 0 0 1 0 0 0
Matrix n = 5 counts the following compositions:
           i=-3:        i=-1:          i=1:            i=3:        i=5:
        -----------------------------------------------------------------
   k=1: |    0            0             0               0          (5)
   k=2: |   (14)         (23)          (32)            (41)         0
   k=3: |    0          (131)       (221)(122)   (311)(113)(212)    0
   k=4: |    0       (1211)(1112)  (2111)(1121)         0           0
   k=5: |    0            0          (11111)            0           0

Gus Wiseman, A raster plot of the zeros in A(16).

The number of nonzero terms in each matrix appears to be A000096.
The number of zeros in each matrix appears to be A000124.
Row sums and column sums both appear to be A007318 (Pascal's triangle).
The matrix sums are A131577.
Antidiagonal sums appear to be A163493.
The reverse-alternating version is also A345197 (this sequence).
Antidiagonals are A345907.
Traces are A345908.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Other tetrangles: A318393, A318816, A320808, A321912.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A007318, A008549, A025047, A032443, A034871, A114121, A120452, A238279, A239830, A344604.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[#]==k&&ats[#]==i&]],{n,0,6},{k,1,n},{i,-n+2,n,2}]

A120452 Number of partitions of n-1 boys and one girl with no couple.

Original entry on oeis.org

1, 1, 3, 5, 9, 14, 23, 34, 52, 75, 109, 153, 216, 296, 407, 549, 739, 981, 1300, 1702, 2224, 2879, 3716, 4761, 6083, 7721, 9774, 12306, 15450, 19307, 24064, 29867, 36978, 45614, 56130, 68846, 84250, 102793, 125148, 151955, 184123, 222553, 268482

Offset: 1

Yasutoshi Kohmoto, Jul 20 2006

From Gus Wiseman, Jun 08 2021: (Start)
Also the number of:
- integer partitions of 2n with reverse-alternating sum 2;
- reversed integer partitions of 2n with alternating sum 2;
- integer partitions of 2n with exactly two odd parts, one of which is the greatest;
- odd-length integer partitions of 2n whose conjugate partition has exactly two odd parts.
Note that integer partitions of 2n with alternating or reverse-alternating sum 0 are counted by A000041, ranked by A000290.
(End)

			n=5:
If partitions have no pair "o*", then a(5)=9 ("o" means a boy, "*" means a girl): {o, o, o, o, *}, {o, o, *, oo}, {*, oo, oo}, {o, *, ooo}, {o, o, oo*}, {oo, oo*}, {*, oooo}, {o, ooo*}, {oooo*}.
From _Gus Wiseman_, Jun 08 2021: (Start)
The a(1) = 1 through a(6) = 14 partitions of 2n with reverse-alternating sum 2:
  (2)  (211)  (222)    (332)      (442)        (552)
              (321)    (431)      (541)        (651)
              (21111)  (22211)    (22222)      (33222)
                       (32111)    (32221)      (33321)
                       (2111111)  (33211)      (43221)
                                  (43111)      (44211)
                                  (2221111)    (54111)
                                  (3211111)    (2222211)
                                  (211111111)  (3222111)
                                               (3321111)
                                               (4311111)
                                               (222111111)
                                               (321111111)
                                               (21111111111)
For example, the partition (43221) has reverse-alternating sum 1 - 2 + 2 - 3 + 4 = 2, so is counted under a(6).
The a(1) = 1 through a(6) = 14 partitions of 2n with exactly two odd parts, one of which is the greatest:
  (11)  (31)  (33)   (53)    (55)     (75)
              (51)   (71)    (73)     (93)
              (321)  (332)   (91)     (111)
                     (521)   (532)    (543)
                     (3221)  (541)    (552)
                             (721)    (732)
                             (3322)   (741)
                             (5221)   (921)
                             (32221)  (5322)
                                      (5421)
                                      (7221)
                                      (33222)
                                      (52221)
                                      (322221)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A000070, A002865.
A diagonal of A103919.
A diagonal of A344612.
A000097 counts partitions of 2n with alternating sum 2.
A001700/A088218 appear to count compositions with reverse-alternating sum 2.
A058696 counts partitions of 2n, ranked by A300061.
A344610 counts partitions of 2n by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A001250, A027187, A119899, A124754, A239830, A316524, A325535, A344618, A344651, A344739.

a[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021 *)

a(n) = A000070(n-2) + A002865(n-1). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006
a(n) = A000070(n-1) - A000041(n-2) = A000070(n-3) + A000041(n-1). - Max Alekseyev, Aug 23 2006
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 - 37*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Oct 25 2016

More terms from Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Aug 15 2006

More terms from Max Alekseyev, Aug 23 2006

A026810 Number of partitions of n in which the greatest part is 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350

Offset: 0

Clark Kimberling

Also number of partitions of n into exactly 4 parts.
Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - R. J. Mathar, Nov 03 2018
From Gus Wiseman, Jun 27 2021: (Start)
Also the number of strict integer partitions of 2n with alternating sum 4, or (by conjugation) partitions of 2n covering an initial interval of positive integers with exactly 4 odd parts. The strict partitions with alternating sum 4 are:
  (4)  (5,1)  (6,2)    (7,3)    (8,4)      (9,5)      (10,6)
              (5,2,1)  (5,3,2)  (5,4,3)    (6,5,3)    (7,6,3)
                       (6,3,1)  (6,4,2)    (7,5,2)    (8,6,2)
                                (7,4,1)    (8,5,1)    (9,6,1)
                                (6,3,2,1)  (6,4,3,1)  (6,5,4,1)
                                           (7,4,2,1)  (7,4,3,2)
                                                      (7,5,3,1)
                                                      (8,5,2,1)
                                                      (6,4,3,2,1)
(End)

			From _Gus Wiseman_, Jun 27 2021: (Start)
The a(4) = 1 through a(10) = 9 partitions of length 4:
  (1111)  (2111)  (2211)  (2221)  (2222)  (3222)  (3322)
                  (3111)  (3211)  (3221)  (3321)  (3331)
                          (4111)  (3311)  (4221)  (4222)
                                  (4211)  (4311)  (4321)
                                  (5111)  (5211)  (4411)
                                          (6111)  (5221)
                                                  (5311)
                                                  (6211)
                                                  (7111)
(End)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
D. E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Washington Bomfim, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Cf. A001400, A026811, A026812, A026813, A026814, A026815, A026816, A069905 (3 positive parts), A002621 (partial sums), A005044 (first differences).
A non-strict version is A000710 or A088218.
This is column k = 2 of A152146.
A reverse version is A343941.
Cf. A000041, A000070, A000097, A067659, A103919, A120452, A236559, A239830, A306145, A343942, A344616, A344649, A344651.

[Round((n^3+3*n^2-9*n*(n mod 2))/144): n in [0..60]]; // Vincenzo Librandi, Oct 14 2015

A049347 := proc(n)
        op(1+(n mod 3),[1,-1,0]) ;
end proc:
A056594 := proc(n)
        op(1+(n mod 4),[1,0,-1,0]) ;
end proc:
A026810 := proc(n)
        1/288*(n+1)*(2*n^2+4*n-13+9*(-1)^n) ;
        %-A049347(n)/9 ;
        %+A056594(n)/8 ;
end proc: # R. J. Mathar, Jul 03 2012

Table[Count[IntegerPartitions[n], {4, _}], {n, 0, 60}]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {0, 0, 0, 0, 1, 1, 2, 3, 5, 6}, 60] (* Vincenzo Librandi, Oct 14 2015 *)
Table[Length[IntegerPartitions[n, {4}]], {n, 0, 60}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^4/Product[1 - x^k, {k, 1, 4}], {x, 0, 60}], x] (* Robert A. Russell, May 13 2018 *)

for(n=0, 60, print(n, " ", round((n^3 + 3*n^2 -9*n*(n % 2))/144))); \\ Washington Bomfim, Jul 03 2012

x='x+O('x^60); concat([0, 0, 0, 0], Vec(x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)))) \\ Altug Alkan, Oct 14 2015

vector(60, n, n--; (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 + real(I^n)/8 - ((n+2)%3-1)/9) \\ Altug Alkan, Oct 26 2015

print1(0,", "); for(n=1,60,j=0;forpart(v=n,j++,,[4,4]); print1(j,", ")) \\ Hugo Pfoertner, Oct 01 2018

G.f.: x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) = x^4/((1-x)^4*(1+x)^2*(1+x+x^2)*(1+x^2)).
a(n+4) = A001400(n). - Michael Somos, Apr 07 2012
a(n) = round( (n^3 + 3*n^2 -9*n*(n mod 2))/144 ). - Washington Bomfim, Jan 06 2021 and Jul 03 2012
a(n) = (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 -A049347(n)/9 +A056594(n)/8. - R. J. Mathar, Jul 03 2012
From Gregory L. Simay, Oct 13 2015: (Start)
a(n) = (n^3 + 3*n^2 - 9*n)/144 + a(m) - (m^3 + 3*m^2 - 9*m)/144 if n = 12k + m and m is odd. For example, a(23) = a(12*1 + 11) = (23^3 + 3*23^2 - 9*23)/144 + a(11) - (11^3 + 3*11^2 - 9*11)/144 = 94.
a(n) = (n^3 + 3*n^2)/144 + a(m) - (m^3 + 3*m^2)/144 if n = 12k + m and m is even. For example, a(22) = a(12*1 + 10) = (22^3 + 3*22^2)/144 + a(10) - (10^3 + 3*10^2)/144 = 84. (End)
a(n) = A008284(n,4). - Robert A. Russell, May 13 2018
From Gregory L. Simay, Jul 28 2019: (Start)
a(2n+1) = a(2n) + a(n+1) - a(n-3) and
a(2n) = a(2n-1) + a(n+2) - a(n-2). (End)

A344608 Number of integer partitions of n with reverse-alternating sum < 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 3, 7, 7, 14, 15, 27, 29, 49, 54, 86, 96, 146, 165, 242, 275, 392, 449, 623, 716, 973, 1123, 1498, 1732, 2274, 2635, 3411, 3955, 5059, 5871, 7427, 8620, 10801, 12536, 15572, 18065, 22267, 25821, 31602, 36617, 44533, 51560, 62338, 72105, 86716

Offset: 0

Gus Wiseman, May 30 2021

nonn

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed of integer partitions of n with alternating sum < 0.
No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.
Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).
A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)    (42)    (43)      (53)      (54)
              (41)    (51)    (52)      (62)      (63)
              (2111)  (3111)  (61)      (71)      (72)
                              (2221)    (3221)    (81)
                              (3211)    (4211)    (3222)
                              (4111)    (5111)    (3321)
                              (211111)  (311111)  (4221)
                                                  (4311)
                                                  (5211)
                                                  (6111)
                                                  (222111)
                                                  (321111)
                                                  (411111)
                                                  (21111111)

The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.
The strict case (for n > 2) is A067659 (odd bisection: A344650).
The Heinz numbers of these partitions are A119899 (complement: A344609).
The bisections are A236914 (odd) and A344743 (even).
The ordered version appears to be A294175 (even bisection: A008549).
The complement is counted by A344607 (even bisection: A344611).
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with alternating sum <= 0, ranked by A028260.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions with rev-alternating sum 2 (negative: A344741).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.

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

A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 24, 26, 33, 34, 35, 38, 39, 40, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 104, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 135, 136, 140, 141, 142, 143, 145, 146, 150

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Also numbers whose alternating sum of prime indices is < 0. Equivalently, numbers with even bigomega whose conjugate prime indices are not all even. This is the intersection of A028260 and A000037. - Gus Wiseman, Jun 20 2021

			From _Gus Wiseman_, Jun 20 2021: (Start)
The sequence of terms together with their prime indices begins:
       6: {1,2}          51: {2,7}          86: {1,14}
      10: {1,3}          54: {1,2,2,2}      87: {2,10}
      14: {1,4}          55: {3,5}          88: {1,1,1,5}
      15: {2,3}          56: {1,1,1,4}      90: {1,2,2,3}
      21: {2,4}          57: {2,8}          91: {4,6}
      22: {1,5}          58: {1,10}         93: {2,11}
      24: {1,1,1,2}      60: {1,1,2,3}      94: {1,15}
      26: {1,6}          62: {1,11}         95: {3,8}
      33: {2,5}          65: {3,6}          96: {1,1,1,1,1,2}
      34: {1,7}          69: {2,9}         104: {1,1,1,6}
      35: {3,4}          74: {1,12}        106: {1,16}
      38: {1,8}          77: {4,5}         111: {2,12}
      39: {2,6}          82: {1,13}        115: {3,9}
      40: {1,1,1,3}      84: {1,1,2,4}     118: {1,17}
      46: {1,9}          85: {3,7}         119: {4,7}
(End)

Superset: A119847. Subset: A006881. The intersection of A028260 and A000037.
Positions of negative terms in A316524.
The partitions with these Heinz numbers are counted by A344608.
Complement of A344609.
Cf. A000041, A000070, A000097, A027187, A103919, A116406, A239829, A239830, A343941, A344607, A344651.

Select[Range[200],And@@EvenQ[{PrimeOmega[#],DivisorSigma[0,#]}]&] (* Harvey P. Dale, Jan 24 2013 *)

A345924 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -2.

Original entry on oeis.org

12, 40, 49, 51, 54, 60, 144, 161, 163, 166, 172, 184, 194, 197, 199, 202, 205, 207, 212, 217, 219, 222, 232, 241, 243, 246, 252, 544, 577, 579, 582, 588, 600, 624, 642, 645, 647, 650, 653, 655, 660, 665, 667, 670, 680, 689, 691, 694, 700, 720, 737, 739, 742

Offset: 1

Gus Wiseman, Jul 11 2021

nonn

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

			The initial terms and the corresponding compositions:
     12: (1,3)          202: (1,3,2,2)        582: (3,4,1,2)
     40: (2,4)          205: (1,3,1,2,1)      588: (3,3,1,3)
     49: (1,4,1)        207: (1,3,1,1,1,1)    600: (3,2,1,4)
     51: (1,3,1,1)      212: (1,2,2,3)        624: (3,1,1,5)
     54: (1,2,1,2)      217: (1,2,1,3,1)      642: (2,6,2)
     60: (1,1,1,3)      219: (1,2,1,2,1,1)    645: (2,5,2,1)
    144: (3,5)          222: (1,2,1,1,1,2)    647: (2,5,1,1,1)
    161: (2,5,1)        232: (1,1,2,4)        650: (2,4,2,2)
    163: (2,4,1,1)      241: (1,1,1,4,1)      653: (2,4,1,2,1)
    166: (2,3,1,2)      243: (1,1,1,3,1,1)    655: (2,4,1,1,1,1)
    172: (2,2,1,3)      246: (1,1,1,2,1,2)    660: (2,3,2,3)
    184: (2,1,1,4)      252: (1,1,1,1,1,3)    665: (2,3,1,3,1)
    194: (1,5,2)        544: (4,6)            667: (2,3,1,2,1,1)
    197: (1,4,2,1)      577: (3,6,1)          670: (2,3,1,1,1,2)
    199: (1,4,1,1,1)    579: (3,5,1,1)        680: (2,2,2,4)

These compositions are counted by A002054.
These are the positions of -2's in A124754.
The version for reverse-alternating sum is A345923.
The opposite (positive 2) version is A345925.
The version for Heinz numbers of partitions is A345962.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A008549, A025047, A163493, A239830, A344609, A344651, A344741, A345908, A345961.

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],ats[stc[#]]==-2&]

cp's OEIS Frontend

A058622 a(n) = 2^(n-1) - ((1+(-1)^n)/4)*binomial(n, n/2).

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A344604 Number of alternating compositions of n, including twins (x,x).

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A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2n with reverse-alternating sum 2k.

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A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

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A345197 Concatenation of square matrices A(n), each read by rows, where A(n)(k,i) is the number of compositions of n of length k with alternating sum i, where 1 <= k <= n, and i ranges from -n + 2 to n in steps of 2.

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A120452 Number of partitions of n-1 boys and one girl with no couple.

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A026810 Number of partitions of n in which the greatest part is 4.

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