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A351637 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct run-lengths using exactly k different symbols, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).

%I A351637 #27 Oct 20 2024 12:39:27
%S A351637 1,0,1,0,1,0,1,2,0,1,2,0,1,4,0,1,10,6,0,1,12,6,0,1,18,12,0,1,26,18,0,
%T A351637 1,56,96,24,0,1,64,102,24,0,1,100,186,48,0,1,132,264,72,0,1,192,420,
%U A351637 120,0,1,350,1344,864,120,0,1,434,1572,936,120
%N A351637 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct run-lengths using exactly k different symbols, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).
%C A351637 Permuting the symbols will not change the structure.
%C A351637 Equivalently, T(n,k) is the number of restricted growth strings [s(0), s(1), ..., s(n-1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1, the maximum value is k and every run has a different length.
%H A351637 Andrew Howroyd, <a href="/A351637/b351637.txt">Table of n, a(n) for n = 0..958</a> (rows 0..100)
%F A351637 T(n,k) = Sum_{j=1..k} R(n,j)*binomial(k, j)*(-1)^(k-j)/k! for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).
%F A351637 T(n,k) = A350824(n,k)/k!.
%F A351637 T(A000217(n),n) = A000142(n). - _Alois P. Heinz_, Feb 15 2022
%e A351637 Triangle begins:
%e A351637   1;
%e A351637   0, 1;
%e A351637   0, 1;
%e A351637   0, 1,   2;
%e A351637   0, 1,   2;
%e A351637   0, 1,   4;
%e A351637   0, 1,  10,   6;
%e A351637   0, 1,  12,   6;
%e A351637   0, 1,  18,  12;
%e A351637   0, 1,  26,  18;
%e A351637   0, 1,  56,  96, 24;
%e A351637   0, 1,  64, 102, 24;
%e A351637   0, 1, 100, 186, 48;
%e A351637   0, 1, 132, 264, 72;
%e A351637   ...
%e A351637 The T(6,1) = 1 word is 111111.
%e A351637 The T(6,2) = 10 words are 111112, 111122, 111211, 111221, 112111, 112221, 112222, 122111, 122211, 122222.
%e A351637 The T(6,3) = 6 words are 111223, 111233, 112333, 112223, 122333, 122233.
%o A351637 (PARI)
%o A351637 P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
%o A351637 R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
%o A351637 T(n)={my(u=P(n), v=concat([1], sum(k=1, n, R(u, k)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/r!) ))); [Vecrev(p) | p<-v]}
%o A351637 { my(A=T(16)); for(n=1, #A, print(A[n])) }
%Y A351637 Row sums are A351638.
%Y A351637 Partial row sums include A000007, A000012, A032020, A351639.
%Y A351637 Column k=2 is A216695.
%Y A351637 Cf. A000142, A000217, A003056, A008289, A350824, A351641.
%K A351637 nonn,tabf
%O A351637 0,8
%A A351637 _Andrew Howroyd_, Feb 15 2022