A027306 -id:A027306 - cp's OEIS frontend

A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

0, 12, 160, 193, 195, 198, 204, 216, 240, 2304, 2561, 2563, 2566, 2572, 2584, 2608, 2656, 2752, 2944, 3074, 3077, 3079, 3082, 3085, 3087, 3092, 3097, 3099, 3102, 3112, 3121, 3123, 3126, 3132, 3152, 3169, 3171, 3174, 3180, 3192, 3232, 3265, 3267, 3270, 3276

Offset: 1

Gus Wiseman, Nov 21 2021

nonn

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 terms and corresponding compositions begin:
       0: ()
      12: (1,3)
     160: (2,6)
     193: (1,6,1)
     195: (1,5,1,1)
     198: (1,4,1,2)
     204: (1,3,1,3)
     216: (1,2,1,4)
     240: (1,1,1,5)
    2304: (3,9)
    2561: (2,9,1)
    2563: (2,8,1,1)
    2566: (2,7,1,2)
    2572: (2,6,1,3)
    2584: (2,5,1,4)

These compositions are counted by A224274 up to 0's.
Except for 0, a subset of A345919.
The positive version is A348614, reverse A349153.
An unordered version is A348617, counted by A001523.
The reverse version is A349155.
A positive unordered version is A349159, counted by A000712 up to 0's.
A000346 = even-length compositions with alt sum != 0, complement A001700.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000984, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917, A349160.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Necklaces are ranked by A065609, dual A333764, reversed A333943.
- Alternating compositions are ranked by A345167, complement A345168.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Total[stc[#]]==-2*ats[stc[#]]&]

A100068 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)3^(n-2k).

Original entry on oeis.org

1, 3, 11, 36, 123, 408, 1370, 4560, 15235, 50760, 169326, 564336, 1881582, 6271632, 20907156, 69689376, 232304355, 774343560, 2581169510, 8603882160, 28679699578, 95598937008, 318663476076, 1062211351776, 3540705857998, 11802351958608, 39341178395660, 131137257852000

Offset: 0

Paul Barry, Nov 02 2004

An inverse Chebyshev transform of x/(1-3*x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, Sum_{k=0..floor(n/2)} binomial(n,k) * r^(n-2*k) has g.f. 2*x/(sqrt(1-4*x^2)*(r*sqrt(1-4*x^2) + 2*x - r)). - corrected by Vaclav Kotesovec, Dec 06 2012

Generally (for r>1), a(n) ~ (r + 1/r)^n. - Vaclav Kotesovec, Dec 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A027306, A100067, A100069.

m:=3; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 08 2022

CoefficientList[Series[2*x/(Sqrt[1-4*x^2]*(3*Sqrt[1-4*x^2] + 2*x-3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 06 2012 *)

my(x='x+O('x^66)); Vec(2*x/(sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x-3))) \\ Joerg Arndt, May 12 2013

m=3; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 08 2022

G.f.: 2*x/(sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x-3)).
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*3^(n-k).
a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1+(-1)^(n-k))*3^k/2.
D-finite with recurrence 9*n*a(n) +12*(-3*n+1)*a(n-1) +4*(-4*n-1)*a(n-2) +48*(3*n-4)*a(n-3) +80*(-n+3)*a(n-4)=0. - R. J. Mathar, Nov 22 2012
a(n) ~ 10^n/3^n. - Vaclav Kotesovec, Dec 06 2012

A100069 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)4^(n-2k).

Original entry on oeis.org

1, 4, 18, 76, 326, 1384, 5892, 25036, 106438, 452344, 1922588, 8170936, 34726940, 147589264, 627256088, 2665837516, 11329815878, 48151714264, 204644809932, 869740430056, 3696396920116, 15709686864304, 66766169526008, 283756220309176, 1205963937666076, 5125346734404784

Offset: 0

Paul Barry, Nov 02 2004

An inverse Chebyshev transform of x/(1-4*x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, Sum_{k=0..floor(n/2)} binomial(n,k) * r^(n-2*k) has g.f. 2*x/(sqrt(1-4*x^2)*(r*sqrt(1-4*x^2) + 2*x - r)). - corrected by Vaclav Kotesovec, Dec 06 2012

Generally (for r>1), a(n) ~ (r + 1/r)^n. - Vaclav Kotesovec, Dec 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A027306, A100067, A100068.

m:=4; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 08 2022

CoefficientList[Series[x/(Sqrt[1-4*x^2]*(2*Sqrt[1-4*x^2]+x-2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 06 2012 *)

my(x='x+O('x^66)); Vec(x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2))) \\ Joerg Arndt, May 12 2013

m=4; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 08 2022

G.f.: x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2)). - corrected by Vaclav Kotesovec, Dec 06 2012
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*4^(n-2*k).
a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1 + (-1)^(n-k))*4^k/2.
8*n*a(n) = 2*(19*n-4)*a(n-1) + (15*n+2)*a(n-2) - 8*(19*n-23)*a(n-3) + 68*(n-3)*a(n-4) = 0. - R. J. Mathar, Nov 22 2012
a(n) ~ 17^n/4^n. - Vaclav Kotesovec, Dec 06 2012

A292548 Number of multisets of nonempty binary words with a total of n letters such that no word has a majority of 0's.

Original entry on oeis.org

1, 1, 4, 8, 25, 53, 148, 328, 858, 1938, 4862, 11066, 27042, 61662, 147774, 336854, 795678, 1810466, 4228330, 9597694, 22211897, 50279985, 115489274, 260686018, 594986149, 1339215285, 3040004744, 6823594396, 15416270130, 34510814918, 77644149076, 173368564396

Offset: 0

Alois P. Heinz, Sep 18 2017

nonn

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 4: {01}, {10}, {11}, {1,1}.
a(3) = 8: {011}, {101}, {110}, {111}, {1,01}, {1,10}, {1,11}, {1,1,1}.

Alois P. Heinz, Table of n, a(n) for n = 0..3213

Row sums of A292506.
Column k=2 of A292712.
Cf. A027306.

g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      g(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

g[n_] :=  2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*
     g[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

G.f.: Product_{j>=1} 1/(1-x^j)^A027306(j).

Euler transform of A027306.

A346632 Triangle read by rows giving the main diagonals of the matrices counting integer compositions by length and alternating sum (A345197).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 6, 6, 0, 0, 0, 1, 2, 9, 12, 0, 0, 0, 0, 1, 2, 12, 18, 10, 0, 0, 0, 0, 1, 2, 15, 24, 30, 20, 0, 0, 0, 0, 1, 2, 18, 30, 60, 60, 0, 0, 0, 0, 0, 1, 2, 21, 36, 100, 120, 35, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 26 2021

The matrices (A345197) count the integer 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. The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			Triangle begins:
   1
   0   0
   0   1   0
   0   1   2   0
   0   1   2   0   0
   0   1   2   3   0   0
   0   1   2   6   6   0   0
   0   1   2   9  12   0   0   0
   0   1   2  12  18  10   0   0   0
   0   1   2  15  24  30  20   0   0   0
   0   1   2  18  30  60  60   0   0   0   0
   0   1   2  21  36 100 120  35   0   0   0   0
   0   1   2  24  42 150 200 140  70   0   0   0   0
   0   1   2  27  48 210 300 350 280   0   0   0   0   0
   0   1   2  30  54 280 420 700 700 126   0   0   0   0   0

The first nonzero element in each column appears to be A001405.
These are the diagonals of the matrices given by A345197.
Antidiagonals of the same matrices are A345907.
Row sums are A345908.
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).
Other diagonals are A008277 of A318393 and A055884 of A320808.
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, A000346, A007318, A008549, A025047, A163493, A344610.

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

A347048 Number of even-length ordered factorizations of n with integer alternating product.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 6, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 11, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 6, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 8, 0, 1, 1, 7, 0, 0, 0, 1, 0

Offset: 1

Gus Wiseman, Oct 10 2021

nonn

An ordered factorization of n is a sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) ordered factorizations for n = 16, 32, 36, 48, 64, 96:
  4*4       8*4       6*6       12*4      8*8           24*4
  8*2       16*2      12*3      24*2      16*4          48*2
  2*2*2*2   2*2*4*2   18*2      2*2*6*2   32*2          3*2*8*2
            4*2*2*2   2*2*3*3   3*2*4*2   2*2*4*4       4*2*6*2
                      2*3*3*2   4*2*3*2   2*2*8*2       6*2*4*2
                      3*2*2*3   6*2*2*2   2*4*4*2       8*2*3*2
                      3*3*2*2             4*2*2*4       12*2*2*2
                                          4*2*4*2       2*2*12*2
                                          4*4*2*2
                                          8*2*2*2
                                          2*2*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 0's are A005117 \ {2}.
The restriction to powers of 2 is A027306.
Heinz numbers of partitions of this type are A028260 /\ A347457.
Positions of 3's appear to be A030514.
Positions of 1's are 1 and A082293.
Allowing non-integer alternating product gives A174725, unordered A339846.
The odd-length version is A347049.
The unordered version is A347438, reverse A347439.
Allowing any length gives A347463.
Partitions of this type are counted by A347704, reverse A035363.
A001055 counts factorizations (strict A045778, ordered A074206).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A339890 counts odd-length factorizations, ordered A174726.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347446 = partitions with integer alternating product, reverse A347445.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A116406, A138364, A347440, A347441, A347454, A347456, A347458, A347459, A347464.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[ordfacs[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

A347048(n, m=n, ap=1, e=0) = if(1==n,!(e%2) && 1==numerator(ap), sumdiv(n, d, if(d>1, A347048(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(n) = A347463(n) - A347049(n).

Data section extended up to a(105) by Antti Karttunen, Jul 28 2024

A348701 a(n) = Sum_{x_1+x_2+x_3+x_4=n, 0 <= x_1 <=x_2 <= x_3 <= x_4 <= n} (3n)!/((n-x_1)! (n-x_2)! * (n-x_3)! * (n-x_4)!).

Original entry on oeis.org

1, 6, 270, 14280, 1027950, 49945896, 3576428856, 206746398144, 14538815734014, 856120482145500, 57732481295380320, 3504640795033499160, 243836930413853027160, 14665028066155264295400, 977424087385506563672160, 61042393779786411602842080

Offset: 0

Seiichi Manyama, Oct 30 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..535 (terms 0..200 from Seiichi Manyama)
Vaclav Kotesovec, Plot of a(n+1)/a(n) for n = 1..500

Cf. A027306, A092473, A348704.

Cf. A348700.

Table[Sum[If[a + b + c + d == n, (3 n)!/((n - a)!*(n - b)!*(n - c)!*(n - d)!), 0], {a, 0, n}, {b, 0, a}, {c, 0, b}, {d, 0, c}], {n, 0, 15}] (* Michael De Vlieger, Oct 30 2021 *)

a(n) = sum(a=0, n, sum(b=a, n, sum(c=b, n, sum(d=c, n, if(a+b+c+d==n, (3*n)!/((n-a)!*(n-b)!*(n-c)!*(n-d)!), 0)))));

A348704 a(n) = Sum_{x_1+x_2+ ... +x_n=n, 0 <= x_1<= x_2 <= ... <= x_n <= n} ((n-1)n)!/((n-x_1)! (n-x_2)! * ... * (n-x_n)!).

Original entry on oeis.org

1, 1, 3, 170, 1027950, 1079901406584, 448687115051986530720, 89290138377185872821028908288000, 14759276773881730859717740767606565269685350000, 2387650794422480788739162652666454048976136433287918499830000000

Offset: 0

Seiichi Manyama, Oct 30 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..15

Cf. A027306, A092473, A348701.

Cf. A005651, A348703.

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def A(k, n)
  sum = 0
  m = f((k - 1) * n)
  (0..n).to_a.repeated_combination(k){|i|
    if (0..k - 1).inject(0){|s, j| s + i[j]} == n
      sum += m / (0..k - 1).inject(1){|s, j| s * f(n - i[j])}
    end
  }
  sum
end
def A348704(n)
  (0..n).map{|i| A(i, i)}
end
p A348704(7)

A349153 Numbers k such that the k-th composition in standard order has sum equal to twice its reverse-alternating sum.

Original entry on oeis.org

0, 11, 12, 14, 133, 138, 143, 148, 155, 158, 160, 168, 179, 182, 188, 195, 198, 204, 208, 216, 227, 230, 236, 240, 248, 2057, 2066, 2071, 2077, 2084, 2091, 2094, 2101, 2106, 2111, 2120, 2131, 2134, 2140, 2149, 2154, 2159, 2164, 2171, 2174, 2192, 2211, 2214

Offset: 1

Gus Wiseman, Nov 17 2021

nonn

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

			The terms and corresponding compositions begin:
    0: ()
   11: (2,1,1)
   12: (1,3)
   14: (1,1,2)
  133: (5,2,1)
  138: (4,2,2)
  143: (4,1,1,1,1)
  148: (3,2,3)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)
  160: (2,6)
  168: (2,2,4)
  179: (2,1,3,1,1)
  182: (2,1,2,1,2)
  188: (2,1,1,1,3)

These compositions are counted by A262977 up to 0's.
Except for 0, a subset of A345917.
The unreversed version is A348614.
The unreversed negative version is A349154.
The negative version is A349155.
A non-reverse unordered version is A349159, counted by A000712 up to 0's.
An unordered version is A349160, counted by A006330 up to 0's.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A025047 counts alternating or wiggly compositions, complement A345192.
A034871, A097805, and A345197 count compositions by alternating sum.
A103919 counts partitions by alternating sum, reverse A344612.
A116406 counts compositions with alternating sum >=0, ranked by A345913.
A138364 counts compositions with alternating sum 0, ranked by A344619.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A294175.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- Heinz number is given by A333219.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz compositions are ranked by A333489, complement A348612.
- Alternating compositions are ranked by A345167, complement A345168.

stc[n_]:=Differences[ Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,1000],Total[stc[#]]==2*sats[stc[#]]&]

A042971 a(n) = (C(2n, n)/2 - (2^(n-1) + ((n+1) mod 2)*C(n-1, n/2-1)))/2.

Original entry on oeis.org

0, 0, 3, 12, 55, 210, 826, 3136, 12027, 45870, 175846, 674784, 2598102, 10024196, 38771188, 150250496, 583368787, 2268706134, 8836184878, 34461323872, 134563944322, 526023515996, 2058355584748, 8061896050432, 31602643220830, 123979613859660, 486734822857596

Offset: 1

Wouter Meeussen

nonn

Number of distinct asymmetric staircase walks connecting opposite corners of a square grid of side n > 1. - Christian Barrientos, Nov 25 2018

			U = up, R = right Example of asymmetric staircase walk: URURUURR. - _Christian Barrientos_, Nov 29 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A027306 (for symmetric staircase walks).

[(Binomial(2*n,n) -(2^n +(1+(-1)^n)*Binomial(n-1, Floor(n/2)-1) ))/4: n in [1..30]]; // G. C. Greubel, Feb 17 2019

Table[Binomial[2n,n]/2-(2^(n-1)+Mod[n+1,2]Binomial[n-1, n/2-1]),{n,30}]/2

a(n) = (binomial(2*n,n)/2 - (2^(n-1) + if (((n+1) % 2), binomial( n-1, n/2-1))))/2; \\ Michel Marcus, Nov 25 2018

[(binomial(2*n,n) -(2^n + (1+(-1)^n)*binomial(n-1, floor(n/2)-1)))/4 for n in (1..30)] # G. C. Greubel, Feb 17 2019

cp's OEIS Frontend

A349154 Numbers k such that the k-th composition in standard order has sum equal to negative twice its alternating sum.

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A100068 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*3^(n-2*k).

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A100069 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*4^(n-2*k).

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A292548 Number of multisets of nonempty binary words with a total of n letters such that no word has a majority of 0's.

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Maple

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A346632 Triangle read by rows giving the main diagonals of the matrices counting integer compositions by length and alternating sum (A345197).

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A347048 Number of even-length ordered factorizations of n with integer alternating product.

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A348701 a(n) = Sum_{x_1+x_2+x_3+x_4=n, 0 <= x_1 <=x_2 <= x_3 <= x_4 <= n} (3*n)!/((n-x_1)! * (n-x_2)! * (n-x_3)! * (n-x_4)!).

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A348704 a(n) = Sum_{x_1+x_2+ ... +x_n=n, 0 <= x_1<= x_2 <= ... <= x_n <= n} ((n-1)*n)!/((n-x_1)! * (n-x_2)! * ... * (n-x_n)!).

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A100068 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)3^(n-2k).

A100069 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)4^(n-2k).

A348701 a(n) = Sum_{x_1+x_2+x_3+x_4=n, 0 <= x_1 <=x_2 <= x_3 <= x_4 <= n} (3n)!/((n-x_1)! (n-x_2)! * (n-x_3)! * (n-x_4)!).

A348704 a(n) = Sum_{x_1+x_2+ ... +x_n=n, 0 <= x_1<= x_2 <= ... <= x_n <= n} ((n-1)n)!/((n-x_1)! (n-x_2)! * ... * (n-x_n)!).