A227310 -id:A227310 - cp's OEIS frontend

A372646 Irregular triangle read by rows, T(n,k) is the number of integer compositions of n such that their set of adjacent differences is a subset of {-1,1}, they contain 1 as a part, and have k parts. T(n,k) for n >= 0, floor(sqrt(2(n+1))-(1/2)) <= k <= floor((2n+1)/3).

0, 1, 0, 2, 0, 1, 0, 1, 2, 2, 0, 0, 1, 0, 4, 1, 0, 0, 3, 2, 2, 2, 0, 1, 0, 3, 6, 1, 0, 2, 0, 4, 2, 0, 2, 8, 3, 0, 1, 0, 0, 0, 6, 8, 1, 2, 8, 5, 0, 5, 2, 0, 0, 7, 14, 4, 0, 1, 0, 4, 6, 0, 10, 10, 1, 0, 0, 8, 20, 8, 0, 6, 2, 0, 2, 3, 0, 14, 22, 5, 0, 1, 0, 0, 6

Offset: 0

John Tyler Rascoe, May 08 2024

Is there a bijection between the unrestricted compositions of k-1 and compositions of this kind with k parts for k > 0?

			T(10,4) = 2: (1,2,3,4), (4,3,2,1).
T(10,5) = 2: (2,1,2,3,2), (2,3,2,1,2).
T(10,7) = 1: (1,2,1,2,1,2,1).
Triangle T(n,k) begins:
  0;
  .  1;
  .  0;
  .  .  2;
  .  .  0, 1;
  .  .  0, 1;
  .  .  .  2, 2;
  .  .  .  0, 0, 1;
  .  .  .  0, 4, 1;
  .  .  .  0, 0, 3, 2;
  .  .  .  .  2, 2, 0, 1;
  ...

John Tyler Rascoe, Rows n = 0..70, flattened
John Tyler Rascoe, Python program.

Cf. A131577 (empirical column sums), A372647 (row sums).

Cf. A003242, A173258, A227310, A309938, A364039.

Python
```
# see linked program
```

G.f. for k-th column is C(x,k) - (x^k)*C(x,k) for k > 0 where C(x,k) is the g.f of the k-th column of A309938.

A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 95, 260, 713, 1959, 5386, 14815, 40759, 112151, 308609, 849240, 2337009, 6431246, 17698332, 48704714, 134032593, 368850417, 1015056867, 2793383746, 7687248186, 21154913043, 58217239536, 160210872557, 440892153268, 1213312738702, 3338974845151, 9188688696438

Offset: 0

Joerg Arndt, Jul 06 2013

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
Cf. A227310 (g.f.: 1/G(0), where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).

nmax = 40; CoefficientList[Series[1/(1 - x/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 2] - Floor[Range[nmax + 2]/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+1) / G(k+1) ) );
gf = 1 /(1- q/G(0));
Vec(gf)

G.f.: 1/(1-q/ (1-q/(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. A(x) = 1/(1 - B(x)) where B(x) is the g.f. of A006958.
a(n) ~ c * d^n, where d = 2.751949072495748078279227332764623096815571855905843246297955690122791154... and c = 0.215973947378529032758849789768859077066690378163074586384819930605436492... - Vaclav Kotesovec, Sep 05 2017

A383620 Number of weak compositions of n such that the set of adjacent differences is a subset of {-1,1}.

Original entry on oeis.org

1, 4, 5, 9, 13, 20, 30, 45, 66, 102, 152, 229, 344, 518, 780, 1180, 1775, 2676, 4037, 6088, 9182, 13852, 20891, 31512, 47536, 71706, 108166, 163172, 246140, 371303, 560118, 844943, 1274606, 1922767, 2900522, 4375493, 6600511, 9956990, 15020307, 22658428

Offset: 0

John Tyler Rascoe, May 02 2025

nonn

			a(0) = 1: (0).
a(1) = 4: (0,1), (0,1,0), (1,0), (1).
...
a(4) = 13: (0,1,0,1,0,1,0,1), (0,1,0,1,0,1,0,1,0), (1,0,1,0,1,0,1,0), (1,0,1,0,1,0,1), (0,1,0,1,2), (1,0,1,2), (2,1,0,1,0), (2,1,0,1), (0,1,2,1,0), (0,1,2,1), (1,2,1,0), (1,2,1), (4).

Alois P. Heinz, Table of n, a(n) for n = 0..5598

Cf. A007318, A173258, A214247, A214249, A227310.

M(k) = matrix(k+1,k+1, i,j, if(i==j,1,if(i==j-1, -x^(i-1), if(i==j+1, -x^(i-1), 0))))
A_x(N) = {my(k=N+1,x='x+O('x^k)); Vec(vecsum(M(k)^(-1) * vector(k+1,i,x^(i-1))~))}
A_x(10)

cp's OEIS Frontend

A372646 Irregular triangle read by rows, T(n,k) is the number of integer compositions of n such that their set of adjacent differences is a subset of {-1,1}, they contain 1 as a part, and have k parts. T(n,k) for n >= 0, floor(sqrt(2(n+1))-(1/2)) <= k <= floor((2n+1)/3).

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A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

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Mathematica

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Formula

A383620 Number of weak compositions of n such that the set of adjacent differences is a subset of {-1,1}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A372646 Irregular triangle read by rows, T(n,k) is the number of integer compositions of n such that their set of adjacent differences is a subset of {-1,1}, they contain 1 as a part, and have k parts. T(n,k) for n >= 0, floor(sqrt(2*(n+1))-(1/2)) <= k <= floor((2*n+1)/3).

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Author

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Comments

Examples

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Programs

Python

Formula

A227045 G.f.: 1/(1 - q/G(0)) where G(k) = 1 - q^(k+1) / (1 - q^(k+1) / G(k+1) ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383620 Number of weak compositions of n such that the set of adjacent differences is a subset of {-1,1}.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A372646 Irregular triangle read by rows, T(n,k) is the number of integer compositions of n such that their set of adjacent differences is a subset of {-1,1}, they contain 1 as a part, and have k parts. T(n,k) for n >= 0, floor(sqrt(2(n+1))-(1/2)) <= k <= floor((2n+1)/3).