A228352 Triangle read by rows, giving antidiagonals of an array of sequences representing the number of compositions of n when there are N types of ones (the sequences in the array begin (1, N, ...)).

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

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

Cf. A073133, A011782, A001519, A007052, A018902, A018903, A018904.

Sequence in context: A160232 A026300 A099514 * A303911 A205575 A368338

Adjacent sequences: A228349 A228350 A228351 * A228353 A228354 A228355

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 20 2013

Status

approved