A175331 -id:A175331 - cp's OEIS frontend

A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 4, 7, 8, 1, 1, 1, 2, 4, 8, 13, 13, 1, 1, 1, 2, 4, 8, 15, 24, 21, 1, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1, 1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 29 2008

Polynomial expansion as antidiagonal of p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)). Based on the Pisot general polynomial type q(x,n) = x^n - (x^n-1)/(x-1) (the original name of the sequence).
Row sums are 1, 2, 3, 5, 8, 14, ... (A079500).
Conjecture: Since the array row sequences successively tend to A000079, the absolute values of nonzero differences between two successive row sequences tend to A045623 = {1,2,5,12,28,64,144,320,704,1536,...}, as k -> infinity. - L. Edson Jeffery, Dec 26 2013

			Array A begins:
  {1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1, ...}
  {1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89, ...}
  {1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274, ...}
  {1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, ...}
  {1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, ...}
  {1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, ...}
  ... - _L. Edson Jeffery_, Dec 26 2013
As a triangle:
  {1},
  {1, 1},
  {1, 1, 1},
  {1, 1, 2, 1},
  {1, 1, 2, 3, 1},
  {1, 1, 2, 4, 5, 1},
  {1, 1, 2, 4, 7, 8, 1},
  {1, 1, 2, 4, 8, 13, 13, 1},
  {1, 1, 2, 4, 8, 15, 24, 21, 1},
  {1, 1, 2, 4, 8, 16, 29, 44, 34, 1},
  {1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1},
  {1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1},
  {1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 125, 236, 401, 504, 233, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 464, 773, 927, 377, 1}

Cf. A000079, A045623, A092921, A175331.
Same as A048887 but with a column of 1's added on the left (the number of compositions of 0 is defined to be equal to 1).
Array rows (with appropriate offsets) are A000012, A000045, A000073, A000078, A001591, A001592, etc.

g[x_, n_] = x^(n) - (x^n - 1)/(x - 1);
h[x_, n_] = FullSimplify[ExpandAll[x^(n)*g[1/x, n]]];
f[t_, n_] := 1/h[t, n];
a = Table[CoefficientList[Series[f[t, m], {t, 0, 30}], t], {m, 1, 31}];
b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}];
Flatten[b] (* Triangle version *)
Grid[Table[CoefficientList[Series[(1 - x)/(1 - 2 x + x^(n + 1)), {x, 0, 10}], x], {n, 1, 10}]] (* Array version - L. Edson Jeffery, Jul 18 2014 *)

t(n,m) = antidiagonal_expansion of p(x,n) where p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)).

G.f. for array A: (1-x)/(1 - 2*x + x^(n+1)), n>=1. - L. Edson Jeffery, Dec 26 2013

Definition changed by L. Edson Jeffery, Jul 18 2014

A245437 Expansion of x^5/(x^6-x^4-x^2-x+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 10, 17, 29, 50, 86, 147, 252, 432, 741, 1270, 2177, 3732, 6398, 10968, 18802, 32232, 55255, 94723, 162382, 278369, 477204, 818064, 1402395, 2404105, 4121322, 7065122, 12111635, 20762798, 35593360, 61017175, 104600848, 179315699

Offset: 0

Vincenzo Librandi, Jul 22 2014

G.f. taken from p. 12 of the Brlek et al. reference.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Srecko Brlek, Andrea Frosini, Simone Rinaldi, Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol. 13 (2006).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,0,-1).

Cf. A079500, A175331, A244521, A245185.

[n le 6 select Floor(n/6) else Self(n-1)+Self(n-2)+Self(n-4)-Self(n-6): n in [1..50]];

CoefficientList[Series[x^5/(x^6 - x^4 - x^2 - x + 1), {x, 0, 50}], x]
LinearRecurrence[{1, 1, 0, 1, 0, -1}, {0, 0, 0, 0, 0, 1}, 50] (* Bruno Berselli, Jul 22 2014 *)

G.f.: x^5/(x^6 - x^4 - x^2 - x + 1).

a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-6) for n>5.

A175353 Antidiagonal expansion of (x + x^(m + 1))/(1 - 2*x - x^(m + 1)).

Original entry on oeis.org

2, 6, 1, 18, 3, 1, 54, 7, 2, 1, 162, 17, 5, 2, 1, 486, 41, 11, 4, 2, 1, 1458, 99, 24, 9, 4, 2, 1, 4374, 239, 53, 19, 8, 4, 2, 1, 13122, 577, 117, 40, 17, 8, 4, 2, 1

Offset: 0

Roger L. Bagula, Dec 03 2010

Row sums are {0, 2, 7, 22, 64, 187, 545, 1597, 4700, 13888, ...};

I reversed the signs on Riordan's Fibonacci function.

			{2},
{6, 1},
{18, 3, 1},
{54, 7, 2, 1},
{162, 17, 5, 2, 1},
{486, 41, 11, 4, 2, 1},
{1458, 99, 24, 9, 4, 2, 1},
{4374, 239, 53, 19, 8, 4, 2, 1},
{13122, 577, 117, 40, 17, 8, 4, 2, 1}

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 125 and 155.

Cf. A175331.

f[x_, n_] = (x + x^(m + 1))/(1 - 2*x - x^(m + 1));
a = Table[Table[SeriesCoefficient[
      Series[f[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 0, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];
Flatten[%]

G.f.: f(x,m) = (x + x^(m + 1))/(1 - 2*x - x^(m + 1)).

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A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

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A245437 Expansion of x^5/(x^6-x^4-x^2-x+1).

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A175353 Antidiagonal expansion of (x + x^(m + 1))/(1 - 2*x - x^(m + 1)).

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