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A048887 Array T read by antidiagonals, where T(m,n) = number of compositions of n into parts <= m. 17
1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 7, 8, 1, 1, 2, 4, 8, 13, 13, 1, 1, 2, 4, 8, 15, 24, 21, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Taking finite differences of array columns from the top down, we obtain (1; 1,1; 1,2,1; 1,4,2,1;...) = A048004 rows. - Gary W. Adamson, Aug 20 2010

T(m,n) is the number of binary words of length n-1 with < m consecutive 1's. - Geoffrey Critzer, Sep 02 2012

REFERENCES

J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978, p. 154.

LINKS

Alois P. Heinz, Antidiagonals n = 1..141, flattened

FORMULA

G.f.: (1-z)/[1-2z+z^(t+1)].

EXAMPLE

T(2,5) counts 11111, 1112, 1121, 1211, 2111, 122, 212, 221, where "1211" abbreviates the composition 1+2+1+1.

These eight compositions correspond respectively to: {0,0,0,0}, {0,0,0,1}, {0,0,1,0}, {0,1,0,0}, {1,0,0,0}, {0,1,0,1}, {1,0,0,1}, {1,0,1,0} per the bijection given by N. J. A. Sloane in A048004. - Geoffrey Critzer, Sep 02 2012

The array begins:

1,1,1,1,1,1,1,...

1,2,3,5,8,13,...

1,2,4,7,13,...

1,2,4,8,...

MAPLE

G := t->(1-z)/(1-2*z+z^(t+1)): T := (m, n)->coeff(series(G(m), z=0, 30), z^n): matrix(7, 12, T);

# second Maple program:

T:= proc(m, n) option remember; `if`(n=0 or m=1, 1,

      add(T(m, n-j), j=1..min(n, m)))

    end:

seq(seq(T(1+d-n, n), n=1..d), d=1..14); # Alois P. Heinz, May 21 2013

MATHEMATICA

Table[nn=10; a=(1-x^k)/(1-x); b=1/(1-x); c=(1-x^(k-1))/(1-x); CoefficientList[ Series[a b/(1-x^2 b c), {x, 0, nn}], x], {k, 1, nn}]//Grid  (* Geoffrey Critzer, Sep 02 2012 *)

T[m_, n_] := T[m, n] = If[n == 0 || m == 1, 1, Sum[T[m, n-j], {j, 1, Min[n, m]}]]; Table[Table[T[1+d-n, n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-Fran├žois Alcover, Nov 12 2014, after Alois P. Heinz *)

CROSSREFS

Rows: A000045 (Fibonacci), A000073 (tribonacci), A000078 (tetranacci), etc.

Essentially a reflected version of A092921. See A048004 and A126198 for closely related arrays.

Cf. A048004. - Gary W. Adamson, Aug 20 2010

Sequence in context: A004070 A180562 A199711 * A047913 A152977 A259799

Adjacent sequences:  A048884 A048885 A048886 * A048888 A048889 A048890

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified August 22 10:37 EDT 2017. Contains 290946 sequences.