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A351645 Triangle read by rows: T(n,k) is the number of length n word structures using exactly k different symbols with all distinct run-lengths and the first run length of a symbol less than that of previous symbols, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).

%I A351645 #18 Oct 20 2024 12:39:16
%S A351645 1,0,1,0,1,0,1,1,0,1,1,0,1,2,0,1,5,1,0,1,6,1,0,1,9,2,0,1,13,3,0,1,28,
%T A351645 16,1,0,1,32,17,1,0,1,50,31,2,0,1,66,44,3,0,1,96,70,5,0,1,175,224,36,
%U A351645 1,0,1,217,262,39,1,0,1,308,428,71,2,0,1,425,619,105,3
%N A351645 Triangle read by rows: T(n,k) is the number of length n word structures using exactly k different symbols with all distinct run-lengths and the first run length of a symbol less than that of previous symbols, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).
%C A351645 Permuting the symbols will not change the structure.
%C A351645 The k-th column of A351637 is divisible by k!.
%H A351645 Andrew Howroyd, <a href="/A351645/b351645.txt">Table of n, a(n) for n = 0..958</a> (rows 0..100)
%F A351645 T(n,k) = Sum_{j=1..k} R(n,j)*binomial(k, j)*(-1)^(k-j)/(k!)^2 for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).
%F A351645 T(n,k) = A351637(n,k)/k! = A350824(n,k)/(k!)^2.
%F A351645 T(A000217(n),n) = 1.
%e A351645 Triangle begins:
%e A351645   1;
%e A351645   0, 1;
%e A351645   0, 1;
%e A351645   0, 1,  1;
%e A351645   0, 1,  1;
%e A351645   0, 1,  2;
%e A351645   0, 1,  5,  1;
%e A351645   0, 1,  6,  1;
%e A351645   0, 1,  9,  2;
%e A351645   0, 1, 13,  3;
%e A351645   0, 1, 28, 16, 1;
%e A351645   0, 1, 32, 17, 1;
%e A351645   0, 1, 50, 31, 2;
%e A351645   0, 1, 66, 44, 3;
%e A351645   0, 1, 96, 70, 5;
%e A351645   ...
%e A351645 The T(8,1) = 1 word is 11111111.
%e A351645 The T(8,2) = 9 words are 11111112, 11111122, 11111211, 11111221, 11111222, 11112111, 11112221, 11121111, 11211111.
%e A351645 The T(8,3) = 2 words are 11111223, 11112223.
%e A351645 In the last example, the word 11111223 corresponds with 6 words in A351637 which are 11111223, 11111233, 11222223, 11233333, 12222233, 12233333.
%o A351645 (PARI)
%o A351645 P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
%o A351645 R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
%o A351645 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!)^2) ))); [Vecrev(p) | p<-v]}
%o A351645 { my(A=T(16)); for(n=1, #A, print(A[n])) }
%Y A351645 Row sums are A351732.
%Y A351645 Cf. A000217, A003056, A008289, A350824, A351637.
%K A351645 nonn,tabf
%O A351645 0,14
%A A351645 _Andrew Howroyd_, Feb 16 2022