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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, 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 (list; table; graph; refs; listen; history; text; internal format)
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 are:

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 (, 6,...). First few rows of the triangle are:

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-Fran├žois 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 A257006

Adjacent sequences:  A228349 A228350 A228351 * A228353 A228354 A228355

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Aug 20 2013

STATUS

approved

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Last modified July 8 01:42 EDT 2020. Contains 335502 sequences. (Running on oeis4.)