A227310 -id:A227310 - cp's OEIS frontend

A173258 Number of compositions of n where differences between neighboring parts are in {-1,1}.

1, 1, 1, 3, 2, 4, 5, 5, 7, 10, 9, 14, 16, 19, 24, 31, 35, 45, 55, 66, 84, 104, 124, 156, 192, 236, 292, 363, 444, 551, 681, 839, 1040, 1287, 1586, 1967, 2430, 3001, 3717, 4597, 5683, 7034, 8697, 10758, 13312, 16469, 20369, 25204, 31180, 38574, 47726, 59047

Offset: 0

Alois P. Heinz, Jul 08 2012

nonn

			a(3) = 3: [3], [2,1], [1,2].
a(4) = 2: [4], [1,2,1].
a(5) = 4: [5], [3,2], [2,3], [2,1,2].
a(6) = 5: [6], [3,2,1], [2,1,2,1], [1,2,3], [1,2,1,2].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
John Tyler Rascoe, Illustration of n = 1..16

Column k=1 of A214247, A214249.
Row sums of A309938, A364039.
Cf. A227310, A291904, A291905, A343795, A362500, A363718, A364529.

b:= proc(n, i) option remember;
      `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j), j=[-1, 1])))
    end:
a:= n-> `if`(n=0, 1, add(b(n, j), j=1..n)):
seq(a(n), n=0..70);

b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j], {j, {-1, 1}}]]]; a[n_] := If[n == 0, 1, Sum[b[n, j], {j, 1, n}]]; Table[a[n], {n, 0, 70}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

step(R,n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + if(j+1<=n, R[i-j, j+1])) )}
a(n)={my(R=matid(n), t=(n==0), m=0); while(R, m++; t+=vecsum(R[n,]); R=step(R,n)); t} \\ Andrew Howroyd, Aug 23 2019

a(n) ~ c * d^n, where d=1.23729141259673487395949649334678514763130846902468..., c=1.134796087242490181499736234755111281606636700030106.... - Vaclav Kotesovec, May 01 2014

G.f.: 1 + Sum_{k>0} G(x,k) where G(x,k) = x^k*(1 + G(x,k+1) + G(x,k-1)) for k > 0 and G(x,0) = 0. - John Tyler Rascoe, Sep 16 2023

A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 19, 34, 63, 115, 213, 391, 723, 1333, 2463, 4547, 8403, 15522, 28686, 53006, 97963, 181042, 334606, 618415, 1142994, 2112545, 3904592, 7216810, 13338856, 24654268, 45568784, 84225393, 155675230, 287737327, 531830605, 982993368, 1816887637, 3358192905

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Sums along falling diagonals of A161492 (skew Ferrers diagrams by area and number of columns). [Joerg Arndt, Mar 23 2014]

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. P. Delest, J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics. vol.112, no.1-3, pp.65-79, (1993)

Cf. A049346 (g.f.: 1-1/G(0), G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227310 (g.f.: 1/G(0), G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).
Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226729 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).

nmax = 40; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[2, nmax] - Floor[Range[2, nmax]/2])]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 05 2017 *)

N = 66;  q = 'q + O('q^N);
G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+2) / G(k+1) ) );
Vec( 1 / G(0) )

/* formula from the Delest/Fedou reference with t=q: */
N=66;  q='q+O('q^N);  t=q;
qn(n) = prod(k=1, n, 1-q^k );
nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) );
dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n );
v=Vec(nm/dn)

G.f.: 1/(1-q /(1-q^2/(1-q^2/(1-q^3/(1-q^3/(1-q^4/(1-q^4/(1-q^5/(1-q^5/(1-...) )) )) )) )) ).
G.f.: 1/x - Q(0)/(2*x), where Q(k)= 1 + 1/(1 - 1/(1 - 1/(2*x^(k+1)) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
G.f.: 1/x - U(0)/x, where U(k)= 1 - x^(k+1)/(1 - x^(k+1)/U(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
G.f.: -W(0)/x, where W(k)= 1 - x^(k+1) - x^k - x^(2*k+2)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
G.f.: G(0) where G(k) = 1 - q/(q^(k+2) - 1 / G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2016
a(n) ~ c * d^n, where d = 1.84832326133106924642685135202616091890310896530577301386219207630312784... and c = 0.244648950328338656997216931963422920467577616734159139510762093105072... - Vaclav Kotesovec, Sep 05 2017

A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 1, 0, 6, 0, 6, 0, 1, 0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1, 0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1, 0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1, 0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1, 0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1

Offset: 0

Joerg Arndt, Mar 29 2014

Triangle A129182 transposed.
Column sums give the Catalan numbers (A000108).
Row sums give A143951.
Sums along falling diagonals give A005169.
T(4n,2n) = A240008(n). - Alois P. Heinz, Mar 30 2014

			Triangle begins:
00:  1;
01:  0, 1;
02:  0, 0, 1;
03:  0, 0, 0, 1;
04:  0, 0, 1, 0, 1;
05:  0, 0, 0, 2, 0, 1;
06:  0, 0, 0, 0, 3, 0, 1;
07:  0, 0, 0, 1, 0, 4, 0, 1;
08:  0, 0, 0, 0, 3, 0, 5, 0, 1;
09:  0, 0, 0, 1, 0, 6, 0, 6, 0, 1;
10:  0, 0, 0, 0, 3, 0, 10, 0, 7, 0, 1;
11:  0, 0, 0, 0, 0, 7, 0, 15, 0, 8, 0, 1;
12:  0, 0, 0, 0, 2, 0, 14, 0, 21, 0, 9, 0, 1;
13:  0, 0, 0, 0, 0, 7, 0, 25, 0, 28, 0, 10, 0, 1;
14:  0, 0, 0, 0, 1, 0, 17, 0, 41, 0, 36, 0, 11, 0, 1;
15:  0, 0, 0, 0, 0, 5, 0, 35, 0, 63, 0, 45, 0, 12, 0, 1;
16:  0, 0, 0, 0, 1, 0, 16, 0, 65, 0, 92, 0, 55, 0, 13, 0, 1;
17:  0, 0, 0, 0, 0, 5, 0, 40, 0, 112, 0, 129, 0, 66, 0, 14, 0, 1;
18:  0, 0, 0, 0, 0, 0, 16, 0, 86, 0, 182, 0, 175, 0, 78, 0, 15, 0, 1;
19:  0, 0, 0, 0, 0, 3, 0, 43, 0, 167, 0, 282, 0, 231, 0, 91, 0, 16, 0, 1;
20:  0, 0, 0, 0, 0, 0, 14, 0, 102, 0, 301, 0, 420, 0, 298, 0, 105, 0, 17, 0, 1;
...
Column k=4 corresponds to the following 14 paths (dots denote zeros):
#:         path              area   steps (Dyck word)
01:  [ . 1 . 1 . 1 . 1 . ]     4     + - + - + - + -
02:  [ . 1 . 1 . 1 2 1 . ]     6     + - + - + + - -
03:  [ . 1 . 1 2 1 . 1 . ]     6     + - + + - - + -
04:  [ . 1 . 1 2 1 2 1 . ]     8     + - + + - + - -
05:  [ . 1 . 1 2 3 2 1 . ]    10     + - + + + - - -
06:  [ . 1 2 1 . 1 . 1 . ]     6     + + - - + - + -
07:  [ . 1 2 1 . 1 2 1 . ]     8     + + - - + + - -
08:  [ . 1 2 1 2 1 . 1 . ]     8     + + - + - - + -
09:  [ . 1 2 1 2 1 2 1 . ]    10     + + - + - + - -
10:  [ . 1 2 1 2 3 2 1 . ]    12     + + - + + - - -
11:  [ . 1 2 3 2 1 . 1 . ]    10     + + + - - - + -
12:  [ . 1 2 3 2 1 2 1 . ]    12     + + + - - + - -
13:  [ . 1 2 3 2 3 2 1 . ]    14     + + + - + - - -
14:  [ . 1 2 3 4 3 2 1 . ]    16     + + + + - - - -
There are no paths with weight < 4, one with weight 4, none with weight 5, 3 with weight 6, etc., therefore column k=4 is
[0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1, 0, 0, 0, ...].
Row n=8 is [0, 0, 0, 0, 3, 0, 5, 0, 1], the corresponding paths of weight=8 are:
Semilength 4:
  [ . 1 . 1 2 1 2 1 . ]
  [ . 1 2 1 . 1 2 1 . ]
  [ . 1 2 1 2 1 . 1 . ]
Semilength 6:
  [ . 1 . 1 . 1 . 1 . 1 2 1 . ]
  [ . 1 . 1 . 1 . 1 2 1 . 1 . ]
  [ . 1 . 1 . 1 2 1 . 1 . 1 . ]
  [ . 1 . 1 2 1 . 1 . 1 . 1 . ]
  [ . 1 2 1 . 1 . 1 . 1 . 1 . ]
Semilength 8:
  [ . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1 . ]

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Sequences obtained by particular choices for x and y in the g.f. F(x,y) are: A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(sqrt(x), sqrt(x))),  A227310 (1+x*F(x, x^2), also 2-1/F(x, 1)), A239928 (F(x^2, x)), A052709 (x*F(1,x+x^2)), A125305 (F(1, x+x^3)), A002212 (F(1, x/(1-x))).
Cf. A047998, A138158, A227543.
Cf. A129181.

b:= proc(x, y, k) option remember;
      `if`(y<0 or y>x or k<0, 0, `if`(x=0, `if`(k=0, 1, 0),
       b(x-1, y-1, k-y+1/2)+ b(x-1, y+1, k-y-1/2)))
    end:
T:= (n, k)-> b(2*k, 0, n):
seq(seq(T(n, k), k=0..n), n=0..20);  # Alois P. Heinz, Mar 29 2014

b[x_, y_, k_] := b[x, y, k] = If[y<0 || y>x || k<0, 0, If[x == 0, If[k == 0, 1, 0], b[x-1, y-1, k-y+1/2] + b[x-1, y+1, k-y-1/2]]]; T[n_, k_] := b[2*k, 0, n]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }
print_triangle(V)= { my( N=#V ); for(n=1, N, print( rvec( V[n]) ) ); }
N=20; x='x+O('x^N);
F(x,y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );
v= Vec( F(x,y) );
print_triangle(v)

G.f.: F(x,y) satisfies F(x,y) = 1 / (1 - x*y * F(x, x^2*y) ).

G.f.: 1/(1 - y*x/(1 - y*x^3/(1 - y*x^5/(1 - y*x^7/(1 - y*x^9/( ... )))))).

A049346 Coefficient of x^(-n) in expansion of continued fraction 0, x, x^2, x^3, x^4, ... .

Original entry on oeis.org

0, 1, 0, 0, -1, 0, 0, 1, 0, 1, -1, 0, -2, 1, -1, 3, -2, 3, -4, 4, -6, 7, -8, 11, -13, 16, -20, 24, -31, 37, -46, 58, -70, 88, -108, 133, -167, 204, -252, 315, -386, 479, -594, 731, -909, 1122, -1386, 1720, -2124, 2628, -3254, 4022, -4980, 6160, -7618, 9432, -11665, 14433, -17860, 22093, -27341, 33824, -41847

Offset: 0

Alain Lasjauniasith (lasjauni(AT)math.u-bordeaux.fr.yyy.com)

sign

Absolute values are essentially A227310. - Franklin T. Adams-Watters, Oct 31 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000

G.f.: 1 - 1/G(0), where G(k)= 1 + x^(k+1)/(1 - x^(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013

G.f.: W(0) - 1, where W(k) = 1 - x^(k+1)/( x^(k+1) - 1/(1 - x^(k+1)/( x^(k+1) + 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 27 2013

Added more terms, Joerg Arndt, Jun 29 2013

A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 5, 5, 9, 11, 15, 23, 28, 43, 57, 78, 113, 149, 214, 293, 403, 569, 774, 1086, 1502, 2072, 2896, 3986, 5548, 7691, 10636, 14797, 20459, 28400, 39386, 54542, 75724, 104886, 145468, 201733, 279545, 387786, 537472, 745233, 1033383, 1432415, 1986394

Offset: 0

Ilya Gutkovskiy, Apr 18 2017

nonn

			G.f.: A(x) = 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 5*x^10 + ...

Robert Israel, Table of n, a(n) for n = 0..5000

Cf. A000040, A206739, A206741, A206742, A206743, A227310, A269353.

R:= 1:
for i from numtheory:-pi(50) to 1 by -1 do
  R:= series(1-x^ithprime(i)/R, x, 51);
od:
R:= series(1/R, x, 51):
seq(coeff(R,x,j),j=0..50); # Robert Israel, Apr 20 2017

nmax = 50; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^Prime[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 1.3864622092472465020397266918102624708859968795203700659786636158522760956... and c = 0.15945087310540003725148530084775272562567007586487061850065597143186... - Vaclav Kotesovec, Aug 25 2017

A291904 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Sep 05 2017

T(n,k) is the number of integer compositions of n with first part 1, last part k, and all adjacent differences in {-1,1}. - John Tyler Rascoe, Aug 14 2023

			First few rows are:
  1;
  0, 1;
  0, 0;
  0, 0, 1;
  0, 1, 0;
  0, 0, 0;
  0, 0, 1, 1;
  0, 1, 0, 0;
  0, 0, 1, 0;
  0, 1, 1, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 1, 0;
  0, 2, 1, 1, 0.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A291905.
Columns 0-1 give A000007, A227310 (for n>0).
Cf. A008289, A173258, A291895, A291929.

T[0, 0] = 1; T[, 0] = 0; T[n?Positive, k_] /; 0 < k <= Floor[(Sqrt[8n+1] - 1)/2] := T[n, k] = T[n-k, k-1] + T[n-k, k+1]; T[, ] = 0;
Table[T[n, k], {n, 0, 20}, {k, 0, Floor[(Sqrt[8n+1] - 1)/2]}] // Flatten (* Jean-François Alcover, May 29 2019 *)

From John Tyler Rascoe, Aug 14 2023: (Start)
This triangle is T_1(n,k) of the general triangle T_m(n,k) for compositions of this kind with first part m.
T_m(n,k) for 0 < m, 0 <= n, and 0 <= k <= A003056(n+A000217(m-1)).
T_m(0,0) = T_m(m,m) = 1.
T_m(n,k) = T_m(n-k,k-1) + T_m(n-k,k+1) for m < n and 0 < k <= A003056(n+A000217(m-1)).
T_m(n,k) = 0 for 0 < n < m or n < k.
T_m(n,0) = 0 for 0 < n. (End)

A238870 Number of compositions of n with c(1) = 1, c(i+1) - c(i) <= 1, and c(i+1) - c(i) != 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 2, 1, 4, 4, 4, 9, 10, 11, 21, 25, 30, 51, 62, 80, 125, 157, 208, 309, 399, 536, 772, 1013, 1373, 1938, 2574, 3503, 4882, 6540, 8918, 12329, 16611, 22672, 31183, 42182, 57588, 78952, 107092, 146202, 200037, 271831, 371057, 507053, 689885, 941558, 1285655, 1750672, 2388951, 3260459, 4442179, 6060948

Offset: 0

Joerg Arndt, Mar 09 2014

nonn

Number of fountains of n coins with at most two successive coins on the same level.

			The a(10) = 4 such compositions are:
:
:   1:  [ 1 2 1 2 1 2 1 ]  (composition)
:
:  o o o
: ooooooo   (rendering as composition)
:
:     O   O   O
:    O O O O O O O  (rendering as fountain of coins)
:
:
:   2:  [ 1 2 1 2 3 1 ]
:
:     o
:  o oo
: oooooo
:
:           O
:      O   O O
:     O O O O O O
:
:
:   3:  [ 1 2 3 1 2 1 ]
:
:   o
:  oo o
: oooooo
:
:       O
:      O O   O
:     O O O O O O
:
:
:   4:  [ 1 2 3 4 ]
:
:    o
:   oo
:  ooo
: oooo
:
:         O
:        O O
:       O O O
:      O O O O
:

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A005169 (fountains of coins), A001524 (weakly unimodal fountains of coins).
Cf. A186085 (1-dimensional sandpiles), A227310 (rough sandpiles).
Cf. A023361 (fountains of coins with all valleys at lowest level).

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      `if`(i=j, 0, b(n-j, j)), j=1..min(n, i+1)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 11 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, b[n-j, j]], {j, 1, Min[n, i+1]}]];
a[n_] := b[n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)

# translation of the Maple program by Alois P. Heinz
@CachedFunction
def F(n, i):
    if n == 0: return 1
    return sum( (i!=j) * F(n-j, j) for j in [1..min(n,i+1)] ) # A238870
#    return sum( F(n-j, j) for j in [1, min(n,i+1)] ) # A005169
def a(n): return F(n, 0)
print([a(n) for n in [0..50]])
# Joerg Arndt, Mar 20 2014

a(n) ~ c / r^n, where r = 0.733216317061133379740342579187365700397652443391231594... and c = 0.172010618097928709454463097802313209201440229976513439... . - Vaclav Kotesovec, Feb 17 2017

A291874 Expansion of 1 - x/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^9/(1 - ... - x^(2*k-1)/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, -1, 0, -1, -1, 0, -2, -1, -1, -3, -2, -3, -4, -4, -6, -7, -8, -11, -13, -16, -20, -24, -31, -37, -46, -58, -70, -88, -108, -133, -167, -204, -252, -315, -386, -479, -594, -731, -909, -1122, -1386, -1720, -2124, -2628, -3254, -4022, -4980

Offset: 0

Seiichi Manyama, Sep 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A049346, A143951, A227310, A291148 (similar sequence).

Convolution inverse of A143951.

a(n) = -A227310(n) for n > 0.

A364039 Triangle read by rows: T(n,k) is the number of integer compositions of n with first part k and differences between neighboring parts in {-1,1}.

Original entry on oeis.org

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

Offset: 0

John Tyler Rascoe, Aug 06 2023

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 1, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 1, 0, 1;
  0, 2, 1, 1, 0, 0, 1;
  0, 1, 1, 1, 1, 0, 0, 1;
  0, 1, 3, 2, 0, 0, 0, 0, 1;
  0, 3, 2, 1, 2, 1, 0, 0, 0, 1;
  0, 2, 3, 2, 1, 0, 0, 0, 0, 0, 1;
  ...
For n = 6 there are a total of 5 compositions:
  T(6,1) = 2: (123), (1212)
  T(6,2) = 1: (2121)
  T(6,3) = 1: (321)
  T(6,6) = 1: (6)

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A291905 (column k=1), A173258 (row sums).

Cf. A227310, A291904, A309938, A362500, A364529.

T:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
     `if`(n=i, 1, add(T(n-i, i+j), j=[-1, 1])))
    end: T(0$2):=1:
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Aug 08 2023

def A364039_rowlist(row_max):
    A = []
    for n in range(0,row_max+1):
        A.append([])
        for k in range(0,n+1):
            z = 0
            if n==k: z += 1
            elif k > 1 and k-1 <= n-k: z += A[n-k][k-1]
            if k+1 <= n-k and k != 0: z += A[n-k][k+1]
            A[n].append(z)
        print(A[n])
A364039_rowlist(12)

T(n,n) = 1.
T(n,k) = T(n-k,k+1) + T(n-k,k-1) for 0 < k < n.
T(n,k) = 0 for n < k.
T(n,0) = 0 for 0 < n.

A239928 Expansion of F(x^2, x) where F(x,y) is the g.f. of A239927.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 1, 1, 4, 3, 2, 5, 6, 4, 6, 10, 8, 9, 15, 15, 15, 22, 26, 26, 33, 43, 45, 52, 69, 76, 85, 109, 127, 141, 173, 209, 235, 278, 340, 390, 452, 550, 643, 742, 890, 1054, 1221, 1445, 1720, 2007, 2356, 2803, 3291, 3853, 4568, 5385, 6309, 7450, 8800, 10330, 12164, 14372, 16905, 19879

Offset: 0

Joerg Arndt, Mar 29 2014

nonn

What does this sequence count?

Joerg Arndt, Table of n, a(n) for n = 0..1000

Cf. A000108 (F(1, x)), A143951 (F(x, 1)), A005169 (F(x, x), with interlaced zeros), A227310 (F(x, x^2)).

N=66; x='x+O('x^N);
F(x, y, d=0)=if (d>N, 1, 1 / (1-x*y * F(x, x^2*y, d+1) ) );
Vec( F(x^2, x) )

G.f.: 1/(1 - x^3/(1 - x^7/(1 - x^11/(1 - x^15/(1 - x^19/(1 - x^23/( ... ))))))).

cp's OEIS Frontend

A173258 Number of compositions of n where differences between neighboring parts are in {-1,1}.

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A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).

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A239927 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength k such that the area between the x-axis and the path is n (n>=0; 0<=k<=n).

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A049346 Coefficient of x^(-n) in expansion of continued fraction 0, x, x^2, x^3, x^4, ... .

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A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

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A291904 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k+1) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A238870 Number of compositions of n with c(1) = 1, c(i+1) - c(i) <= 1, and c(i+1) - c(i) != 0.

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Sage