A103919 -id:A103919 - cp's OEIS frontend

A116406 Expansion of ((1 + x - 2x^2) + (1+x)*sqrt(1-4x^2))/(2(1-4x^2)).

1, 1, 2, 3, 7, 11, 26, 42, 99, 163, 382, 638, 1486, 2510, 5812, 9908, 22819, 39203, 89846, 155382, 354522, 616666, 1401292, 2449868, 5546382, 9740686, 21977516, 38754732, 87167164, 154276028, 345994216, 614429672, 1374282019, 2448023843

Offset: 0

Paul Barry, Feb 13 2006

Interleaving of A114121 and A032443. Row sums of A116405. Binomial transform is A116409.
Appears to be the number of n-digit binary numbers not having more zeros than ones; equivalently, the number of unrestricted Dyck paths of length n not going below the axis. - Ralf Stephan, Mar 25 2008
From Gus Wiseman, Jun 20 2021: (Start)
Also the number compositions of n with alternating sum >= 0, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. The a(0) = 1 through a(5) = 11 compositions are:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (21)   (22)    (32)
                 (111)  (31)    (41)
                        (112)   (113)
                        (121)   (122)
                        (211)   (212)
                        (1111)  (221)
                                (311)
                                (1121)
                                (2111)
                                (11111)
(End)
From J. Stauduhar, Jan 14 2022: (Start)
Also, for n >= 2, first differences of partial row sums of Pascal's triangle. The first ceiling(n/2)+1 elements of rows n=0 to n=4 in Pascal's triangle are:
     1
    1 1
   1 2
  1 3 3
1 4 6
...
The cumulative sums of these partial rows form the sequence 1,3,6,13,24,..., and its first differences are a(2),a(3),a(4),... in this sequence.
(End)

The alternating sum = 0 case is A001700 or A088218.
The alternating sum > 0 case appears to be A027306.
The bisections are  A032443 (odd) and A114121 (even).
The alternating sum <= 0 version is A058622.
The alternating sum < 0 version is A294175.
The restriction to reversed partitions is A344607.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives the alternating sum of standard compositions.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344616 lists the alternating sums of partitions by Heinz number.
Cf. A000041, A000070, A000097, A003242, A006330, A028260, A058696, A119899, A239830, A344605, A344611, A344650, A344739.

CoefficientList[Series[((1+x-2x^2)+(1+x)Sqrt[1-4x^2])/(2(1-4x^2)),{x,0,40}],x] (* Harvey P. Dale, Aug 16 2012 *)
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, Jun 20 2021 *)

a(n) = A114121(n/2)*(1+(-1)^n)/2 + A032443((n-1)/2)*(1-(-1)^n)/2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-1,k). - Paul Barry, Oct 06 2007
Conjecture: n*(n-3)*a(n) +2*(-n^2+4*n-2)*a(n-1) -4*(n-2)^2*a(n-2) +8*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 28 2014
a(n) ~ 2^(n-2) * (1 + (3+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, May 30 2016
a(n) = 2^(n-1) - A294175(n). - Gus Wiseman, Jun 27 2021

A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 1, 3, 5, 2, 1, 7, 5, 2, 1, 5, 9, 5, 2, 1, 12, 10, 5, 2, 1, 7, 17, 10, 5, 2, 1, 19, 19, 10, 5, 2, 1, 11, 28, 20, 10, 5, 2, 1, 30, 33, 20, 10, 5, 2, 1, 15, 47, 35, 20, 10, 5, 2, 1, 45, 57, 36, 20, 10, 5, 2, 1, 22, 73, 62, 36, 20, 10, 5, 2, 1

Offset: 0

Gus Wiseman, Jun 05 2021

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.

Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - Gus Wiseman, Nov 29 2021

			Triangle begins:
   1
   1
   1   1
   2   1
   2   2   1
   4   2   1
   3   5   2   1
   7   5   2   1
   5   9   5   2   1
  12  10   5   2   1
   7  17  10   5   2   1
  19  19  10   5   2   1
  11  28  20  10   5   2   1
  30  33  20  10   5   2   1
  15  47  35  20  10   5   2   1
  45  57  36  20  10   5   2   1
  22  73  62  36  20  10   5   2   1
  67  92  64  36  20  10   5   2   1
  30 114 102  65  36  20  10   5   2   1
  97 147 107  65  36  20  10   5   2   1
Row n = 10 counts the following partitions (A = 10):
  (55)          (64)         (73)       (82)     (91)   (A)
  (3322)        (442)        (433)      (622)    (811)
  (4411)        (541)        (532)      (721)
  (222211)      (3331)       (631)      (7111)
  (331111)      (4222)       (5221)     (61111)
  (22111111)    (4321)       (6211)
  (1111111111)  (5311)       (42211)
                (22222)      (52111)
                (32221)      (511111)
                (33211)      (4111111)
                (43111)
                (322111)
                (421111)
                (2221111)
                (3211111)
                (31111111)
                (211111111)
The conjugate version is:
  (A)      (55)      (3331)     (331111)    (31111111)   (1111111111)
  (64)     (73)      (5311)     (511111)    (211111111)
  (82)     (91)      (7111)     (3211111)
  (442)    (433)     (33211)    (4111111)
  (622)    (532)     (43111)    (22111111)
  (4222)   (541)     (52111)
  (22222)  (631)     (61111)
           (721)     (322111)
           (811)     (421111)
           (3322)    (2221111)
           (4321)
           (4411)
           (5221)
           (6211)
           (32221)
           (42211)
           (222211)

This is A103919 with all zeros removed.
The strict version is A152146 interleaved with A152157.
The rows are those of A239830 interleaved with those of A239829.
The reverse version is the right half of A344612.
The strict reverse version is the right half of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with rev-alternating sum <= 0, ranked by A028260.
A124754 lists alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344607 counts partitions with rev-alternating sum >= 0, ranked by A344609.
A344608 counts partitions with rev-alternating sum < 0, ranked by A119899.
A344610 counts partitions of n by positive rev-alternating sum.
A344611 counts partitions of 2n with rev-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
A346697 gives the sum of odd-indexed prime indices (reverse: A346699).
A346702 represents the odd bisection of compositions, sums A209281.
Cf. A000070, A000097, A003242, A006330, A025047, A097805, A114121, A116406, A131577, A344617, A344649, A344650, A344654, A346633.

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

A344654 Number of integer partitions of n of which every permutation has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 7, 11, 16, 20, 28, 37, 50, 65, 84, 106, 140, 175, 222, 277, 350, 432, 539, 663, 819, 999, 1225, 1489, 1816, 2192, 2653, 3191, 3846, 4603, 5516, 6578, 7852, 9327, 11083, 13120, 15532, 18328, 21620, 25430, 29904, 35071, 41110, 48080

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no wiggly permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).

			The a(3) = 1 through a(9) = 11 partitions:
  (111)  (1111)  (2111)   (222)     (2221)     (2222)      (333)
                 (11111)  (3111)    (4111)     (5111)      (3222)
                          (21111)   (31111)    (41111)     (6111)
                          (111111)  (211111)   (221111)    (22221)
                                    (1111111)  (311111)    (51111)
                                               (2111111)   (321111)
                                               (11111111)  (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

Joseph Likar, Table of n, a(n) for n = 0..1000

The Heinz numbers of these partitions are A344653, complement A344742.
The complement is counted by A344740.
The normal case starts 0, 0, 0, then becomes A345162, complement A345163.
Allowing twins (x,x) gives A345165, ranked by A345171.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime indices with twins.
A344614 counts compositions with no consecutive strictly monotone triple.
A345164 counts wiggly permutations of prime indices.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A000041, A000070, A102726, A103919, A333489, A335126, A344607, A344615, A345166, A345168, A345169.

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

a(26)-a(32) from Robert Price, Jun 22 2021

a(33) onwards from Joseph Likar, Sep 06 2023

A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.

Original entry on oeis.org

0, 1, 2, 0, 3, -1, 1, 1, 4, -2, 0, 2, 2, 0, 2, 0, 5, -3, -1, 3, 1, 1, 3, -1, 3, -1, 1, 1, 3, -1, 1, 1, 6, -4, -2, 4, 0, 2, 4, -2, 2, 0, 2, 0, 4, -2, 0, 2, 4, -2, 0, 2, 2, 0, 2, 0, 4, -2, 0, 2, 2, 0, 2, 0, 7, -5, -3, 5, -1, 3, 5, -3, 1, 1, 3, -1, 5, -3, -1, 3

Offset: 0

Gus Wiseman, Jun 03 2021

Up to sign, same as A124754.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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 nonnegative integers together with the corresponding standard compositions and their reverse-alternating sums begins:
  0:     () ->  0    15: (1111) ->  0    30:  (1112) ->  1
  1:    (1) ->  1    16:    (5) ->  5    31: (11111) ->  1
  2:    (2) ->  2    17:   (41) -> -3    32:     (6) ->  6
  3:   (11) ->  0    18:   (32) -> -1    33:    (51) -> -4
  4:    (3) ->  3    19:  (311) ->  3    34:    (42) -> -2
  5:   (21) -> -1    20:   (23) ->  1    35:   (411) ->  4
  6:   (12) ->  1    21:  (221) ->  1    36:    (33) ->  0
  7:  (111) ->  1    22:  (212) ->  3    37:   (321) ->  2
  8:    (4) ->  4    23: (2111) -> -1    38:   (312) ->  4
  9:   (31) -> -2    24:   (14) ->  3    39:  (3111) -> -2
  10:  (22) ->  0    25:  (131) -> -1    40:    (24) ->  2
  11: (211) ->  2    26:  (122) ->  1    41:   (231) ->  0
  12:  (13) ->  2    27: (1211) ->  1    42:   (222) ->  2
  13: (121) ->  0    28:  (113) ->  3    43:  (2211) ->  0
  14: (112) ->  2    29: (1121) -> -1    44:   (213) ->  4
Triangle begins (row lengths A011782):
  0
  1
  2  0
  3 -1  1  1
  4 -2  0  2  2  0  2  0
  5 -3 -1  3  1  1  3 -1  3 -1  1  1  3 -1  1  1

Up to sign, same as the reverse version A124754.
The version for Heinz numbers of partitions is A344616.
Positions of zeros are A344619.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A116406 counts compositions with alternating sum >= 0.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
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, A028260, A119899, A239830, A344605, A344607, A344608, A344650, A344739.

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

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

Original entry on oeis.org

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

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&]

A345165 Number of integer partitions of n without an alternating permutation.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 17, 20, 29, 37, 51, 65, 85, 106, 141, 175, 223, 277, 351, 432, 540, 663, 820, 999, 1226, 1489, 1817, 2192, 2654, 3191, 3847, 4603, 5517, 6578, 7853, 9327, 11084, 13120, 15533, 18328, 21621, 25430, 29905, 35071, 41111, 48080, 56206, 65554, 76420, 88918

Offset: 0

Gus Wiseman, Jun 12 2021

nonn

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)        (333)
               (1111)  (11111)  (222)     (4111)     (2222)      (3222)
                                (3111)    (31111)    (5111)      (6111)
                                (21111)   (211111)   (41111)     (22221)
                                (111111)  (1111111)  (221111)    (51111)
                                                     (311111)    (321111)
                                                     (2111111)   (411111)
                                                     (11111111)  (2211111)
                                                                 (3111111)
                                                                 (21111111)
                                                                 (111111111)

Joseph Likar, Table of n, a(n) for n = 0..1000
Joseph Likar, Java Implementation using QBinomials

Excluding twins (x,x) gives A344654, complement A344740.
The normal case is A345162, complement A345163.
The complement is counted by A345170, ranked by A345172.
The Heinz numbers of these partitions are A345171.
The version for factorizations is A348380, complement A348379.
A version for ordered factorizations is A348613, complement A348610.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345192 counts non-alternating compositions, without twins A348377.
Cf. A000070, A025048, A025049, A103919, A335126, A344605, A344607, A344615, A344653, A345166, A345167, A345168, A345169, A347706, A348609.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],wigQ]=={}&]],{n,0,15}]

a(26) onwards by Joseph Likar, Aug 21 2023

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}]

A344611 Number of integer partitions of 2n with reverse-alternating sum >= 0.

Original entry on oeis.org

1, 2, 4, 8, 15, 27, 48, 81, 135, 220, 352, 553, 859, 1313, 1986, 2969, 4394, 6439, 9357, 13479, 19273, 27353, 38558, 53998, 75168, 104022, 143172, 196021, 267051, 362086, 488733, 656802, 879026, 1171747, 1555997, 2058663, 2714133, 3566122, 4670256, 6096924, 7935184

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 integer partitions of 2n with alternating sum >= 0.
The reverse-alternating sum of a partition is equal to (-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 partitions of 2n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of partitions of 2n whose parts are all even or whose greatest part is odd.

			The a(0) = 1 through a(4) = 15 partitions:
  ()  (2)   (4)     (6)       (8)
      (11)  (22)    (33)      (44)
            (211)   (222)     (332)
            (1111)  (321)     (422)
                    (411)     (431)
                    (2211)    (521)
                    (21111)   (611)
                    (111111)  (2222)
                              (3311)
                              (22211)
                              (32111)
                              (41111)
                              (221111)
                              (2111111)
                              (11111111)

The non-reversed version is A058696 (partitions of 2n).
The ordered version appears to be A114121.
Odd bisection of A344607.
Row sums of A344610.
The strict case is A344650.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A000097 counts partitions with alternating sum 2.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions of 2n with reverse-alternating sum 2.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344612 counts partitions by sum and rev-alt sum (strict: A344739).
A344618 gives reverse-alternating sums of standard compositions.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A001250, A027187, A028260, A116406, A119899, A124754, A152146, A239829, A344608, A344609, A344649, A344651, A344654.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]>=0&]],{n,0,30,2}]

Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...

More terms from Bert Dobbelaere, Jun 12 2021

cp's OEIS Frontend

A116406 Expansion of ((1 + x - 2x^2) + (1+x)*sqrt(1-4x^2))/(2(1-4x^2)).

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A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.

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A344654 Number of integer partitions of n of which every permutation has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

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A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.

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

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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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A345165 Number of integer partitions of n without an alternating permutation.

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