%I #33 Jan 10 2024 16:01:37
%S 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,
%T 1,7,37,136,314,396,233,64,1,8,50,229,712,1351,1352,610,128,1,9,65,
%U 358,1418,3728,5813,4616,1597,256,1,10,82,529,2564,8781,19520,25012,15760,4181,512
%N 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, ...)).
%C 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.
%H Alois P. Heinz, <a href="/A228352/b228352.txt">Rows n = 1..141, flattened</a>
%F Antidiagonals of an array in which a(n+2) = (N+1)*a(n+1) - (n-1)*a(n); with array sequences beginning (1, N, ...).
%F Array sequence beginning (1, N, ...) is the binomial transform of the sequence in A073133 beginning (1, (N-1), ...).
%F Given the first sequence of the array is (1, 1, 2, 4, 8, 16, ...), successive sequences are INVERT transforms of previous sequences.
%F 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).
%e Array sequence beginning (1, 3, 10, 34, 116, ...) is the binomial transform of (1, 2, 5, 12, 70, ...) in A073133.
%e First few sequences in the array:
%e 1, 1, 2, 4, 8, 16, ...; = A011782
%e 1, 2, 5, 13, 34, 89, ...; = A001519
%e 1, 3, 10, 34, 116, 396, ...; = A007052
%e ... followed by A018902, A018903, A018904, the latter beginning (1, 6, ...). First few rows of the triangle:
%e 1;
%e 1, 1;
%e 1, 2, 2;
%e 1, 3, 5, 4;
%e 1, 4, 10, 13, 8;
%e 1, 5, 17, 34, 34, 16;
%e 1, 6, 26, 73, 116, 89, 32;
%e 1, 7, 37, 136, 314, 396, 233, 64;
%e 1, 8, 50, 229, 712, 1351, 1352, 610, 128;
%e 1, 9, 65, 358, 1418, 3728, 5813, 4616, 1597, 256;
%e 1, 10, 82, 529, 2564, 8781, 19520, 25012, 15760, 4181, 512;
%e ...
%p A:= proc(N, n) option remember;
%p `if`(n=0, 1, N*A(N, n-1) +add(A(N, n-j), j=2..n))
%p end:
%p seq(seq(A(d-n, n), n=0..d-1), d=1..11); # _Alois P. Heinz_, Aug 20 2013
%t 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_ *)
%Y Cf. A073133, A011782, A001519, A007052, A018902, A018903, A018904.
%K nonn,tabl
%O 1,5
%A _Gary W. Adamson_, Aug 20 2013