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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, ...)).

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