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A305962 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and fixed first element; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A305962 #19 May 28 2019 08:14:38
%S A305962 1,1,1,1,1,1,1,1,2,1,1,1,3,5,1,1,1,4,12,15,1,1,1,5,22,59,52,1,1,1,6,
%T A305962 35,150,339,203,1,1,1,7,51,305,1200,2210,877,1,1,1,8,70,541,3125,
%U A305962 10922,16033,4140,1,1,1,9,92,875,6756,36479,110844,127643,21147,1
%N A305962 Number A(n,k) of length-n restricted growth strings (RGS) with growth <= k and fixed first element; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A305962 A(n,k) counts strings [s_1, ..., s_n] with 1 = s_1 <= s_i <= k + max_{j<i} s_j.
%H A305962 Alois P. Heinz, <a href="/A305962/b305962.txt">Antidiagonals n = 0..150, flattened</a>
%F A305962 A(n,k) = (n-1)! * [x^(n-1)] exp(x+Sum_{j=1..k} (exp(j*x)-1)/j) for n>0, A(0,k) = 1.
%e A305962 A(0,2) = 1: the empty string.
%e A305962 A(1,2) = 1: 1.
%e A305962 A(2,2) = 3: 11, 12, 13.
%e A305962 A(3,2) = 12: 111, 112, 113, 121, 122, 123, 124, 131, 132, 133, 134, 135.
%e A305962 Square array A(n,k) begins:
%e A305962   1,   1,     1,      1,      1,       1,       1,       1, ...
%e A305962   1,   1,     1,      1,      1,       1,       1,       1, ...
%e A305962   1,   2,     3,      4,      5,       6,       7,       8, ...
%e A305962   1,   5,    12,     22,     35,      51,      70,      92, ...
%e A305962   1,  15,    59,    150,    305,     541,     875,    1324, ...
%e A305962   1,  52,   339,   1200,   3125,    6756,   12887,   22464, ...
%e A305962   1, 203,  2210,  10922,  36479,   96205,  216552,  435044, ...
%e A305962   1, 877, 16033, 110844, 475295, 1530025, 4065775, 9416240, ...
%p A305962 b:= proc(n, k, m) option remember; `if`(n=0, 1,
%p A305962       add(b(n-1, k, max(m, j)), j=1..m+k))
%p A305962     end:
%p A305962 A:= (n, k)-> b(n, k, 1-k):
%p A305962 seq(seq(A(n, d-n), n=0..d), d=0..12);
%p A305962 # second Maple program:
%p A305962 A:= (n, k)-> `if`(n=0, 1, (n-1)!*coeff(series(exp(x+add(
%p A305962               (exp(j*x)-1)/j, j=1..k)), x, n), x, n-1)):
%p A305962 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A305962 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 A305962 A[n_, k_] := b[n, k, 1-k];
%t A305962 Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 27 2019, after _Alois P. Heinz_ *)
%Y A305962 Columns k=0-10 give: A000012, A000110, A080337, A189845, A305964, A305965, A305966, A305967, A305968, A305969, A305970.
%Y A305962 Main diagonal gives: A305963.
%Y A305962 Antidiagonal sums give: A305971.
%Y A305962 Cf. A306024.
%K A305962 nonn,tabl
%O A305962 0,9
%A A305962 _Alois P. Heinz_, Jun 15 2018