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

Offset

1,9

Comments

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

Links

Table of n, a(n) for n=1..89.

Formula

p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n);

t(n,m) = antidiagonal_expansion(p(x,n)).

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

Example

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}

Mathematica

Clear[f, b, a, g, h, n, t]; 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]; Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {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 *)

Crossrefs

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.

Sequence in context: A205573 A119338 A054124 * A238888 A179748 A096670

Adjacent sequences: A144403 A144404 A144405 * A144407 A144408 A144409

Keyword

nonn,tabl

Author

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

Extensions

Definition changed by L. Edson Jeffery, Jul 18 2014

Status

approved