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A306024 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and first element in [k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A306024 #17 May 27 2019 09:08:24
%S A306024 1,1,0,1,1,0,1,2,2,0,1,3,7,5,0,1,4,15,31,15,0,1,5,26,95,164,52,0,1,6,
%T A306024 40,214,717,999,203,0,1,7,57,405,2096,6221,6841,877,0,1,8,77,685,4875,
%U A306024 23578,60619,51790,4140,0,1,9,100,1071,9780,67354,297692,652595,428131,21147,0
%N A306024 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and first element in [k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A306024 A(n,k) counts strings [s_1, ..., s_n] with 1 <= s_i <= k + max(0, max_{j<i} s_j).
%H A306024 Alois P. Heinz, <a href="/A306024/b306024.txt">Antidiagonals n = 0..150, flattened</a>
%F A306024 E.g.f. of column k: exp(Sum_{j=1..k} (exp(j*x)-1)/j).
%e A306024 A(2,3) = 15: 11, 12, 13, 14, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 36.
%e A306024 A(4,1) = 15: 1111, 1112, 1121, 1122, 1123, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1234.
%e A306024 Square array A(n,k) begins:
%e A306024   1,   1,    1,     1,      1,       1,       1,       1, ...
%e A306024   0,   1,    2,     3,      4,       5,       6,       7, ...
%e A306024   0,   2,    7,    15,     26,      40,      57,      77, ...
%e A306024   0,   5,   31,    95,    214,     405,     685,    1071, ...
%e A306024   0,  15,  164,   717,   2096,    4875,    9780,   17689, ...
%e A306024   0,  52,  999,  6221,  23578,   67354,  160201,  335083, ...
%e A306024   0, 203, 6841, 60619, 297692, 1044045, 2943277, 7117789, ...
%p A306024 b:= proc(n, k, m) option remember; `if`(n=0, 1,
%p A306024       add(b(n-1, k, max(m, j)), j=1..m+k))
%p A306024     end:
%p A306024 A:= (n, k)-> b(n, k, 0):
%p A306024 seq(seq(A(n, d-n), n=0..d), d=0..12);
%p A306024 # second Maple program:
%p A306024 A:= (n, k)-> n!*coeff(series(exp(add(
%p A306024     (exp(j*x)-1)/j, j=1..k)), x, n+1), x, n):
%p A306024 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A306024 b[n_, k_, m_] := b[n, k, m] = If[n==0, 1, Sum[b[n-1, k, Max[m, j]], {j, 1, m+k}]];
%t A306024 A[n_, k_] := b[n, k, 0];
%t A306024 Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 27 2019, after _Alois P. Heinz_ *)
%Y A306024 Columns k=0-10 give: A000007, A000110, A002872, A306027, A306028, A306029, A306030, A306031, A306032, A306033, A306034.
%Y A306024 Main diagonal gives A306025.
%Y A306024 Antidiagonal sums give A306026.
%Y A306024 Cf. A305962.
%K A306024 nonn,tabl
%O A306024 0,8
%A A306024 _Alois P. Heinz_, Jun 17 2018