The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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((sqrtint(8*n+1)+1)/2). 6
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, 1, 56, 96, 24, 0, 1, 64, 102, 24, 0, 1, 100, 186, 48, 0, 1, 132, 264, 72, 0, 1, 192, 420, 120, 0, 1, 350, 1344, 864, 120, 0, 1, 434, 1572, 936, 120 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Permuting the symbols will not change the structure.
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.
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! for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).
T(n,k) = A350824(n,k)/k!.
T(A000217(n),n) = A000142(n). - Alois P. Heinz, Feb 15 2022
EXAMPLE
Triangle begins:
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, 1, 56, 96, 24;
0, 1, 64, 102, 24;
0, 1, 100, 186, 48;
0, 1, 132, 264, 72;
...
The T(6,1) = 1 word is 111111.
The T(6,2) = 10 words are 111112, 111122, 111211, 111221, 112111, 112221, 112222, 122111, 122211, 122222.
The T(6,3) = 6 words are 111223, 111233, 112333, 112223, 122333, 122233.
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!) ))); [Vecrev(p) | p<-v]}
{ my(A=T(16)); for(n=1, #A, print(A[n])) }
CROSSREFS
Row sums are A351638.
Partial row sums include A000007, A000012, A032020, A351639.
Column k=2 is A216695.
Sequence in context: A280317 A283304 A058685 * A029300 A096397 A337547
KEYWORD
nonn,tabf
AUTHOR
Andrew Howroyd, Feb 15 2022
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 20 23:09 EDT 2024. Contains 372720 sequences. (Running on oeis4.)