A350824 Triangle read by rows: T(n,k) is the number of patterns of length n with all distinct run lengths and maximum value k, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).

1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 4, 0, 1, 8, 0, 1, 20, 36, 0, 1, 24, 36, 0, 1, 36, 72, 0, 1, 52, 108, 0, 1, 112, 576, 576, 0, 1, 128, 612, 576, 0, 1, 200, 1116, 1152, 0, 1, 264, 1584, 1728, 0, 1, 384, 2520, 2880, 0, 1, 700, 8064, 20736, 14400, 0, 1, 868, 9432, 22464, 14400

Offset: 0

Author

Andrew Howroyd, Feb 12 2022

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1;
  0, 1,   4;
  0, 1,   4;
  0, 1,   8;
  0, 1,  20,   36;
  0, 1,  24,   36;
  0, 1,  36,   72;
  0, 1,  52,  108;
  0, 1, 112,  576,  576;
  0, 1, 128,  612,  576;
  0, 1, 200, 1116, 1152;
  ...
The T(5,1) = 1 pattern is 11111.
The T(5,2) = 8 patterns are 12222, 11222, 11122, 11112, 21111, 22111, 22211, 22221.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..958 (rows 0..100)

Crossrefs

Row sums are A351292.

Cf. A000217, A001044, A003056, A008289, A351637, A351640.

Programs

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)) ))); [Vecrev(p) | p<-v]}
{ my(A=T(16)); for(n=1, #A, print(A[n])) }

Formula

T(n,k) = Sum_{j=1..k} R(n,j)*binomial(k, j)*(-1)^(k-j) for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).

T(n,k) = k! * A351637(n,k).

T(A000217(n),n) = A001044(n). - Alois P. Heinz, Feb 15 2022