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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((sqrtint(8*n+1)+1)/2). 2
 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, 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, 1, 0, 1, 217, 262, 39, 1, 0, 1, 308, 428, 71, 2, 0, 1, 425, 619, 105, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,14 COMMENTS Permuting the symbols will not change the structure. The k-th column of A351637 is divisible by k!. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..958 (rows 0..100) FORMULA 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). T(n,k) = A351637(n,k)/k! = A350824(n,k)/(k!)^2. T(A000217(n),n) = 1. EXAMPLE Triangle begins: 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, 16, 1; 0, 1, 32, 17, 1; 0, 1, 50, 31, 2; 0, 1, 66, 44, 3; 0, 1, 96, 70, 5; ... The T(8,1) = 1 word is 11111111. The T(8,2) = 9 words are 11111112, 11111122, 11111211, 11111221, 11111222, 11112111, 11112221, 11121111, 11211111. The T(8,3) = 2 words are 11111223, 11112223. In the last example, the word 11111223 corresponds with 6 words in A351637 which are 11111223, 11111233, 11222223, 11233333, 12222233, 12233333. PROG (PARI) P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))} R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]} 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]} { my(A=T(16)); for(n=1, #A, print(A[n])) } CROSSREFS Row sums are A351732. Cf. A000217, A003056, A008289, A350824, A351637. Sequence in context: A285212 A262948 A193471 * A182931 A260615 A293298 Adjacent sequences: A351642 A351643 A351644 * A351646 A351647 A351648 KEYWORD nonn,tabf AUTHOR Andrew Howroyd, Feb 16 2022 STATUS approved

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Last modified February 24 07:07 EST 2024. Contains 370294 sequences. (Running on oeis4.)