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A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.

%I A143983 #23 Jan 06 2022 14:25:56
%S A143983 1,2,1,5,1,1,15,2,1,1,52,5,1,1,1,203,13,2,1,1,1,877,36,6,1,1,1,1,4140,
%T A143983 109,17,2,1,1,1,1,21147,359,44,7,1,1,1,1,1,115975,1266,112,23,2,1,1,1,
%U A143983 1,1,678570,4731,304,65,8,1,1,1,1,1,1,4213597,18657,918,165,30,2,1,1,1,1,1,1
%N A143983 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k has a_k(0)=1, followed by (k-1)-fold 0 and a_k(n) shifts k places down under binomial transform.
%C A143983 The matrix inverse starts:
%C A143983      1;
%C A143983     -2,   1;
%C A143983     -3,  -1,  1;
%C A143983     -8,  -1, -1,  1;
%C A143983    -31,  -3,  0, -1,  1;
%C A143983   -132,  -7, -1,  0, -1,  1;
%C A143983   -616, -19, -4,  0,  0, -1, 1; - _R. J. Mathar_, Mar 22 2013
%H A143983 Alois P. Heinz, <a href="/A143983/b143983.txt">Rows n = 1..141, flattened</a>
%H A143983 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A143983 T(n,k) = Sum_{j=0..n-k} C(n-k,j)*T(j,k) if n>=k, else T(n,k) = 1 if n=1, else T(n,k) = 0.
%e A143983 T(5,2) = 5, because [1,3,3,1] * [1,0,1,1] = 5.
%e A143983 Triangle begins:
%e A143983 :      1;
%e A143983 :      2,    1;
%e A143983 :      5,    1,   1;
%e A143983 :     15,    2,   1,  1;
%e A143983 :     52,    5,   1,  1, 1;
%e A143983 :    203,   13,   2,  1, 1, 1;
%e A143983 :    877,   36,   6,  1, 1, 1, 1;
%e A143983 :   4140,  109,  17,  2, 1, 1, 1, 1;
%e A143983 :  21147,  359,  44,  7, 1, 1, 1, 1, 1;
%e A143983 : 115975, 1266, 112, 23, 2, 1, 1, 1, 1, 1;
%p A143983 T:= proc(n, k) option remember; `if`(n<k, `if`(n=0, 1, 0),
%p A143983        add(binomial(n-k, j) *T(j,k), j=0..n-k))
%p A143983     end:
%p A143983 seq(seq(T(n, k), k=1..n), n=1..14);
%t A143983 t[n_, k_] := t[n, k] = If[n < k, If[n == 0, 1, 0], Sum[Binomial[n-k, j]*t[j, k], {j, 0, n-k}]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 13}] // Flatten (* _Jean-François Alcover_, Dec 18 2013, translated from Maple *)
%Y A143983 Columns 1-6 give: A000110, A000994, A000996, A010748, A010749, A010750.
%Y A143983 Cf. A007318.
%K A143983 eigen,nonn,tabl
%O A143983 1,2
%A A143983 _Alois P. Heinz_, Sep 06 2008