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

Author

Gary W. Adamson, Aug 20 2013

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

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

Programs

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 *)

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).