OFFSET
1,5
COMMENTS
The array sequence beginning (1, N, ...) is such that a(n) in the sequence represents the numbers of compositions of n when there are N types of ones.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
FORMULA
Antidiagonals of an array in which a(n+2) = (N+1)*a(n+1) - (n-1)*a(n); with array sequences beginning (1, N, ...).
Array sequence beginning (1, N, ...) is the binomial transform of the sequence in A073133 beginning (1, (N-1), ...).
Given the first sequence of the array is (1, 1, 2, 4, 8, 16, ...), successive sequences are INVERT transforms of previous sequences.
Array sequence beginning (1, N, ...) is such that a(n), n>1 is N*(a) + a(n-1) + a(n-2) + a(n-3) + a(n-4) + ... + a(0).
EXAMPLE
Array sequence beginning (1, 3, 10, 34, 116, ...) is the binomial transform of (1, 2, 5, 12, 70, ...) in A073133.
First few sequences in the array:
1, 1, 2, 4, 8, 16, ...; = A011782
1, 2, 5, 13, 34, 89, ...; = A001519
1, 3, 10, 34, 116, 396, ...; = A007052
... followed by A018902, A018903, A018904, the latter beginning (1, 6, ...). First few rows of the triangle:
1;
1, 1;
1, 2, 2;
1, 3, 5, 4;
1, 4, 10, 13, 8;
1, 5, 17, 34, 34, 16;
1, 6, 26, 73, 116, 89, 32;
1, 7, 37, 136, 314, 396, 233, 64;
1, 8, 50, 229, 712, 1351, 1352, 610, 128;
1, 9, 65, 358, 1418, 3728, 5813, 4616, 1597, 256;
1, 10, 82, 529, 2564, 8781, 19520, 25012, 15760, 4181, 512;
...
MAPLE
A:= proc(N, n) option remember;
`if`(n=0, 1, N*A(N, n-1) +add(A(N, n-j), j=2..n))
end:
seq(seq(A(d-n, n), n=0..d-1), d=1..11); # Alois P. Heinz, Aug 20 2013
MATHEMATICA
A[k_, n_] := A[k, n] = If[n == 0, 1, k*A[k, n-1] + Sum[A[k, n-j], {j, 2, n}]]; Table[A[d-n, n], {d, 1, 11}, {n, 0, d-1}] // Flatten (* Jean-François Alcover, May 27 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 20 2013
STATUS
approved