A027306 -id:A027306 - cp's OEIS frontend

A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.

0, 3, 6, 10, 12, 13, 15, 20, 24, 25, 27, 30, 36, 40, 41, 43, 46, 48, 49, 50, 51, 53, 54, 55, 58, 60, 61, 63, 72, 80, 81, 83, 86, 92, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 111, 116, 120, 121, 123, 126, 136, 144, 145, 147, 150, 156, 160, 161, 162, 163

Offset: 1

Gus Wiseman, Jul 08 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)
     6: (1,2)
    10: (2,2)
    12: (1,3)
    13: (1,2,1)
    15: (1,1,1,1)
    20: (2,3)
    24: (1,4)
    25: (1,3,1)
    27: (1,2,1,1)
    30: (1,1,1,2)
    36: (3,3)
    40: (2,4)
    41: (2,3,1)

The version for Heinz numbers of partitions is A028260 (counted by A027187).
These compositions are counted by A058622.
These are the positions of terms <= 0 in A124754.
The reverse-alternating version is A345916.
The opposite (k >= 0) version is A345917.
The strictly negative (k < 0) version is A345919.
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).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
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, A032443, A114121, A163493, A344607, A344609, A344610.

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[#]]<=0&]

A345916 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum <= 0.

Original entry on oeis.org

0, 3, 5, 9, 10, 13, 15, 17, 18, 23, 25, 29, 33, 34, 36, 39, 41, 43, 45, 46, 49, 50, 53, 55, 57, 58, 61, 63, 65, 66, 68, 71, 75, 77, 78, 81, 85, 89, 90, 95, 97, 98, 103, 105, 109, 113, 114, 119, 121, 125, 129, 130, 132, 135, 136, 139, 141, 142, 145, 147, 149

Offset: 1

Gus Wiseman, Jul 08 2021

nonn

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 terms together with the corresponding compositions begins:
     0: ()
     3: (1,1)
     5: (2,1)
     9: (3,1)
    10: (2,2)
    13: (1,2,1)
    15: (1,1,1,1)
    17: (4,1)
    18: (3,2)
    23: (2,1,1,1)
    25: (1,3,1)
    29: (1,1,2,1)
    33: (5,1)
    34: (4,2)
    36: (3,3)

The version for Heinz numbers of partitions is A000290.
These compositions are counted by A058622.
These are the positions of terms <= 0 in A344618.
The opposite (k >= 0) version is A345914.
The version for unreversed alternating sum is A345915.
The strictly negative (k < 0) version is A345920.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
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, A000346, A008549, A025047, A027187, A028260, A032443, A114121, A163493, A344607, A344610, A345908.

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[0,100],sats[stc[#]]<=0&]

A027914 T(n,0) + T(n,1) + ... + T(n,n), T given by A027907.

Original entry on oeis.org

1, 2, 6, 17, 50, 147, 435, 1290, 3834, 11411, 34001, 101400, 302615, 903632, 2699598, 8068257, 24121674, 72137547, 215786649, 645629160, 1932081885, 5782851966, 17311097568, 51828203475, 155188936431, 464732722872

Offset: 0

Clark Kimberling

Let b(n)=a(n) mod 2; then b(n)=1/2+(-1)^n*(1/2-A010060(floor(n/2))). - Benoit Cloitre, Mar 23 2004
Binomial transform of A027306. Inverse binomial transform of = A032443. Hankel transform is {1, 2, 3, 4, ..., n, ...}. - Philippe Deléham, Jul 20 2005
Sums of rows of the triangle in A111808. - Reinhard Zumkeller, Aug 17 2005
Number of 3-ary words of length n in which the number of 1's does not exceed the number of 0's. - David Scambler, Aug 14 2012
The Gauss congruences a(n*p^k) == a(n^p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. - Peter Bala, Jan 07 2022

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Moa Apagodu and Ji-Cai Liu, Congruence properties for the trinomial coefficients, arXiv:1907.13547 [math.NT], 2019.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 24.

Cf. A025191, A027915, A081673, A092255, A055217, A005773.

a027914 n = sum $ take (n + 1) $ a027907_row n
-- Reinhard Zumkeller, Jan 22 2013

a := n -> simplify((3^n + GegenbauerC(n,-n,-1/2))/2):
seq(a(n), n=0..25); # Peter Luschny, May 12 2016

CoefficientList[ Series[ (1 + x + Sqrt[1 - 2x - 3x^2])/(2 - 4x - 6x^2), {x, 0, 26}], x] (* Robert G. Wilson v, Jul 21 2015 *)
Table[(3^n + Hypergeometric2F1[1/2 - n/2, -n/2, 1, 4])/2, {n, 0, 20}] (* Vladimir Reshetnikov, May 07 2016 *)
f[n_] := Plus @@ Take[ CoefficientList[ Sum[x^k, {k, 0, 2}]^n, x], n +1]; Array[f, 26, 0] (* Robert G. Wilson v, Jan 30 2017 *)

a(n)=sum(i=0,n,polcoeff((1+x+x^2)^n,i,x))

a(n)=sum(i=0,n,sum(j=0,n,sum(k=0,j,if(i+j+k-n,0,(n!/i!/j!/k!)))))

x='x+O('x^99); Vec((1+x+(1-2*x-3*x^2)^(1/2))/(2*(1-2*x-3*x^2))) \\ Altug Alkan, May 12 2016

a(n) = ( 3^n + A002426(n) )/2; lim n -> infinity a(n+1)/a(n) = 3; 3^n < 2*a(n) < 3^(n+1). - Benoit Cloitre, Sep 28 2002
From Benoit Cloitre, Jan 26 2003: (Start)
a(n) = (1/2) *( Sum{k = 0..n} binomial(n,k)*binomial(n-k,k) + 3^n );
a(n) = Sum_{k = 0..n} Sum_{i = 0..k} binomial(n,i)*binomial(n-i,k);
a(n) = 3^n/2*(1+c/sqrt(n)+O(n^-1/2)) where c=0.5... (End)
c = sqrt(3/Pi)/2 = 0.4886025119... - Vaclav Kotesovec, May 07 2016
a(n) = n!*Sum(i+j+k=n, 1/(i!*j!*k!)) 0<=i<=n, 0<=k<=j<=n. - Benoit Cloitre, Mar 23 2004
G.f.: (1+x+sqrt(1-2x-3x^2))/(2(1-2x-3x^2)); a(n) = Sum_{k = 0..n} floor((k+2)/2)*Sum_{i = 0..floor((n-k)/2)} C(n,i)*C(n-i,i+k)* ((k+1)/(i+k+1)). - Paul Barry, Sep 23 2005; corrected Jan 20 2008
D-finite with recurrence: n*a(n) +(-5*n+4)*a(n-1) +3*(n-2)*a(n-2) +9*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
G.f.: (1+x+1/G(0))/(2*(1-2*x-3*x^2)), where G(k)= 1 + x*(2+3*x)*(4*k+1)/(4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2013
From Peter Bala, Jul 21 2015: (Start)
a(n) = [x^n]( 3*x - 1/(1 - x) )^n.
1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + ... is the o.g.f. for A005773. (End)
a(n) = (3^n + GegenbauerC(n,-n,-1/2))/2. - Peter Luschny, May 12 2016

A038208 Triangle whose (i,j)-th entry is binomial(i,j)*2^i.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 24, 24, 8, 16, 64, 96, 64, 16, 32, 160, 320, 320, 160, 32, 64, 384, 960, 1280, 960, 384, 64, 128, 896, 2688, 4480, 4480, 2688, 896, 128, 256, 2048, 7168, 14336, 17920, 14336, 7168, 2048, 256, 512, 4608, 18432, 43008, 64512, 64512, 43008, 18432, 4608, 512

Offset: 0

N. J. A. Sloane

Triangle obtained from expansion of (2 + 2*x)^n.

			    1;
    2,    2;
    4,    8,    4;
    8,   24,   24,     8;
   16,   64,   96,    64,    16;
   32,  160,  320,   320,   160,   32;
   64,  384,  960,  1280,   960,   384,   64;
  128,  896, 2688,  4480,  4480,  2688,  896,  128;
  256, 2048, 7168, 14336, 17920, 14336, 7168, 2048, 256;

Harvey P. Dale, Table of n, a(n) for n = 0..1000
B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
S. Elizalde, E. Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, corollary 2.2 at m=2, then dropping each 2nd row
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Cf. A000079, A000302 (row sums), A002605 (diagonal sums), A027306.

[Binomial(n,k)*2^n: k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 17 2018

nn=8;Map[Select[#,#>0&]&,Transpose[Table[Range[0,nn]!CoefficientList[Series[2^k x^k/k! Exp[2x],{x,0,nn}],x],{k,0,nn}]]]//Grid (* Geoffrey Critzer, Feb 13 2014 *)
Flatten[Table[Binomial[i,j]2^i,{i,0,10},{j,0,i}]] (* Harvey P. Dale, May 28 2015 *)

for(n=0,10, for(k=0,n, print1(binomial(n,k)*2^n, ", "))) \\ G. C. Greubel, Oct 17 2018

flatten([[binomial(n,k)*2^n for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

E.g.f. for column k: 2^k*x^k/k!*exp(2*x). - Geoffrey Critzer, Feb 13 2014
From G. C. Greubel, Mar 21 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n).
Sum_{k=0..n} T(n, k) = A000302(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A002605(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = 2^n*A027306(n). (End)

A034868 Left half of Pascal's triangle.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 6, 1, 5, 10, 1, 6, 15, 20, 1, 7, 21, 35, 1, 8, 28, 56, 70, 1, 9, 36, 84, 126, 1, 10, 45, 120, 210, 252, 1, 11, 55, 165, 330, 462, 1, 12, 66, 220, 495, 792, 924, 1, 13, 78, 286, 715, 1287, 1716, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 1, 15

Offset: 0

N. J. A. Sloane

			1;
1;
1, 2;
1, 3;
1, 4,  6;
1, 5, 10;
1, 6, 15, 20;
...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A107430, A062344, A122366, A027306 (row sums).
Cf. A008619.
Cf. A225860.
Cf. A126257.
Cf. A034869 (right half), A014413, A014462, A265848.

a034868 n k = a034868_tabf !! n !! k
a034868_row n = a034868_tabf !! n
a034868_tabf = map reverse a034869_tabf
-- Reinhard Zumkeller, improved Dec 20 2015, Jul 27 2012

Flatten[ Table[ Binomial[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}]] (* Robert G. Wilson v, May 28 2005 *)

for(n=0, 14, for(k=0, floor(n/2), print1(binomial(n, k),", ");); print();) \\ Indranil Ghosh, Mar 31 2017

import math
from sympy import binomial
for n in range(15):
    print([binomial(n, k) for k in range(int(math.floor(n/2)) + 1)]) # Indranil Ghosh, Mar 31 2017

from itertools import count, islice
def A034868_gen(): # generator of terms
    yield from (s:=(1,))
    for i in count(0):
        yield from (s:=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + ((s[-1]<<1,) if i&1 else ()))
A034868_list = list(islice(A034868_gen(),30)) # Chai Wah Wu, Oct 17 2023

T(n,k) = A034869(n,floor(n/2)-k), k = 0..floor(n/2). - Reinhard Zumkeller, Jul 27 2012

A292506 Number T(n,k) of multisets of exactly k nonempty binary words with a total of n letters such that no word has a majority of 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 4, 3, 1, 0, 11, 10, 3, 1, 0, 16, 23, 10, 3, 1, 0, 42, 59, 33, 10, 3, 1, 0, 64, 134, 83, 33, 10, 3, 1, 0, 163, 320, 230, 98, 33, 10, 3, 1, 0, 256, 699, 568, 270, 98, 33, 10, 3, 1, 0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1, 0, 1024, 3434, 3439, 1935, 798, 291, 98, 33, 10, 3, 1

Offset: 0

Alois P. Heinz, Sep 17 2017

			T(4,2) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,    1;
  0,   4,    3,    1;
  0,  11,   10,    3,   1;
  0,  16,   23,   10,   3,   1;
  0,  42,   59,   33,  10,   3,  1;
  0,  64,  134,   83,  33,  10,  3,  1;
  0, 163,  320,  230,  98,  33, 10,  3,  1;
  0, 256,  699,  568, 270,  98, 33, 10,  3, 1;
  0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for triangles generated by the Multiset Transformation

Columns k=0-10 give: A000007, A027306 (for n>0), A316403, A316404, A316405, A316406, A316407, A316408, A316409, A316410, A316411.
Row sums give A292548.
T(2n,n) gives A292549.
Cf. A209406, A226873, A290222.

g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);

g[n_] := 2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, x^n, Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1]*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 06 2018, from Maple *)

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

A345908 Traces of the matrices (A345197) counting integer compositions by length and alternating sum.

Original entry on oeis.org

1, 1, 0, 1, 3, 3, 6, 15, 24, 43, 92, 171, 315, 629, 1218, 2313, 4523, 8835, 17076, 33299, 65169

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. Here, the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. So a(n) is the number of compositions of n of length (n + s)/2, where s is the alternating sum of the composition.

			The a(0) = 1 through a(7) = 15 compositions of n = 0..7 of length (n + s)/2 where s = alternating sum (empty column indicated by dot):
  ()  (1)  .  (2,1)  (2,2)    (2,3)    (2,4)      (2,5)
                     (1,1,2)  (1,2,2)  (1,3,2)    (1,4,2)
                     (2,1,1)  (2,2,1)  (2,3,1)    (2,4,1)
                                       (1,1,3,1)  (1,1,3,2)
                                       (2,1,2,1)  (1,2,3,1)
                                       (3,1,1,1)  (2,1,2,2)
                                                  (2,2,2,1)
                                                  (3,1,1,2)
                                                  (3,2,1,1)
                                                  (1,1,1,1,3)
                                                  (1,1,2,1,2)
                                                  (1,1,3,1,1)
                                                  (2,1,1,1,2)
                                                  (2,1,2,1,1)
                                                  (3,1,1,1,1)

Traces of the matrices given by A345197.
Diagonals and antidiagonals of the same matrices are A346632 and A345907.
Row sums of A346632.
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, A001405, A007318, A008549, A025047, A114121, A163493.

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

A246437 Expansion of (1/2)(1/(x+1)+1/(sqrt(-3x^2-2*x+1))).

Original entry on oeis.org

1, 0, 2, 3, 10, 25, 71, 196, 554, 1569, 4477, 12826, 36895, 106470, 308114, 893803, 2598314, 7567465, 22076405, 64498426, 188689685, 552675364, 1620567764, 4756614061, 13974168191, 41088418150, 120906613076, 356035078101, 1049120176954

Offset: 0

Vladimir Kruchinin, Nov 14 2014

a(2101) has 1001 decimal digits. - Michael De Vlieger, Apr 25 2016
This is the analog for Coxeter type B of Motzkin sums (A005043) for Coxeter type A, see the article by Athanasiadis and Savvidou. - F. Chapoton, Jul 20 2017
Number of compositions of n into exactly n nonnegative parts avoiding part 1. a(4) = 10: 0004, 0022, 0040, 0202, 0220, 0400, 2002, 2020, 2200, 4000. - Alois P. Heinz, Aug 19 2018
From Peter Bala, Jan 07 2022: (Start)
The binary transform is A088218. The inverse binomial transform is a signed version of A027306 and the second inverse binomial transform is a signed version of A027914.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..2100
Christos A. Athanasiadis and Christina Savvidou, The Local h-Vector of the Cluster Subdivision of a Simplex, Séminaire Lotharingien de Combinatoire 66 (2012), Article B66c.
Eric Marberg, On some actions of the 0-Hecke monoids of affine symmetric groups, arXiv:1709.07996 [math.CO], 2017.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 24.

Cf. A002426, A005043, A027306, A027914, A088218, A104507, A370616, A386548.

CoefficientList[Series[(1/2) (1 / (x + 1) + 1 / (Sqrt[-3 x^2 - 2 x + 1])), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
Table[(-1)^n (Hypergeometric2F1[1/2, -n, 1, 4] + 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)
Table[Sum[Binomial[n, k] Binomial[n - k - 1, n - 2 k], {k, 0, n/2}], {n, 0, 28}] (* Michael De Vlieger, Apr 25 2016 *)

a(n):=sum(binomial(n,k)*binomial(n-k-1,n-2*k),k,0,n/2);

def a(n):
    if n < 3: return [1,0,2][n]
    return n*hypergeometric([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4)
[simplify(a(n)) for n in (0..28)] # Peter Luschny, Nov 14 2014

a(n) = Sum_{k = 0..n/2} binomial(n,k)*binomial(n-k-1,n-2*k).
A(x) = 1 + x*B'(x)/B(x), where B(x) = (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)) is the o.g.f. of A005043.
a(n) = n*hypergeom([1-n, 1-n/2, 3/2-n/2],[2, 2-n], 4) for n>=3. - Peter Luschny, Nov 14 2014
a(n) ~ 3^(n+1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 14 2014
a(n) = (-1)^n*(hypergeom([1/2, -n], [1], 4) + 1)/2. - Vladimir Reshetnikov, Apr 25 2016
D-finite with recurrence: n*(a(n) - a(n-1)) = (5*n-6)*a(n-2) + 3*(n-2)*a(n-3). - Vladimir Reshetnikov, Oct 13 2016
a(n) = [x^n]( (1 - x + x^2)/(1 - x) )^n. - Peter Bala, Jan 07 2022

A348614 Numbers k such that the k-th composition in standard order has sum equal to twice its alternating sum.

Original entry on oeis.org

0, 9, 11, 14, 130, 133, 135, 138, 141, 143, 148, 153, 155, 158, 168, 177, 179, 182, 188, 208, 225, 227, 230, 236, 248, 2052, 2057, 2059, 2062, 2066, 2069, 2071, 2074, 2077, 2079, 2084, 2089, 2091, 2094, 2098, 2101, 2103, 2106, 2109, 2111, 2120, 2129, 2131

Offset: 1

Gus Wiseman, Oct 29 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 together with their binary indices begin:
    0: ()
    9: (3,1)
   11: (2,1,1)
   14: (1,1,2)
  130: (6,2)
  133: (5,2,1)
  135: (5,1,1,1)
  138: (4,2,2)
  141: (4,1,2,1)
  143: (4,1,1,1,1)
  148: (3,2,3)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)

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

The unordered case (partitions) is counted by A000712, reverse A006330.
These compositions are counted by A262977.
Except for 0, a subset of A345917 (which is itself a subset of A345913).
A000346 = even-length compositions with alt sum != 0, complement A001700.
A011782 counts compositions.
A025047 counts wiggly compositions, ranked by A345167.
A034871 counts compositions of 2n with alternating sum 2k.
A097805 counts compositions by alternating (or reverse-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.
A345197 counts compositions by length and alternating sum.
Cf. A000984, A002458, A004331, A005810, A224274.
Cf. A008549, A013777, A027306, A058622, A088218, A114121, A120452, A126869, A163493, A294175, A344604.

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[#]]&]

A292549 Number of multisets of exactly n nonempty binary words with a total of 2n letters such that no word has a majority of 0's.

Original entry on oeis.org

1, 3, 10, 33, 98, 291, 826, 2320, 6342, 17188, 45750, 120733, 314690, 813854, 2085363, 5306878, 13406382, 33665476, 84031608, 208655086, 515469203, 1267600993, 3103490884, 7567559622, 18381579206, 44487740012, 107301636460, 257967350824, 618279370985

Offset: 0

Alois P. Heinz, Sep 18 2017

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..3163

Cf. A292506.

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+1), 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+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 11 2019, after Alois P. Heinz *)

a(n) = A292506(2n,n) = A292506(2n+j,n+j) for j >= 0.
G.f.: Product_{j>=1} 1/(1-x^j)^A027306(j+1).
Euler transform of j-> A027306(j+1).

cp's OEIS Frontend

A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.

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A345916 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum <= 0.

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A027914 T(n,0) + T(n,1) + ... + T(n,n), T given by A027907.

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A038208 Triangle whose (i,j)-th entry is binomial(i,j)*2^i.

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A034868 Left half of Pascal's triangle.

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A292506 Number T(n,k) of multisets of exactly k nonempty binary words with a total of n letters such that no word has a majority of 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A345908 Traces of the matrices (A345197) counting integer compositions by length and alternating sum.

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A246437 Expansion of (1/2)(1/(x+1)+1/(sqrt(-3x^2-2*x+1))).