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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).

%I A350824 #21 Oct 20 2024 12:39:37
%S A350824 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,
%T A350824 0,1,112,576,576,0,1,128,612,576,0,1,200,1116,1152,0,1,264,1584,1728,
%U A350824 0,1,384,2520,2880,0,1,700,8064,20736,14400,0,1,868,9432,22464,14400
%N 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).
%H A350824 Andrew Howroyd, <a href="/A350824/b350824.txt">Table of n, a(n) for n = 0..958</a> (rows 0..100)
%F A350824 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).
%F A350824 T(n,k) = k! * A351637(n,k).
%F A350824 T(A000217(n),n) = A001044(n). - _Alois P. Heinz_, Feb 15 2022
%e A350824 Triangle begins:
%e A350824   1;
%e A350824   0, 1;
%e A350824   0, 1;
%e A350824   0, 1,   4;
%e A350824   0, 1,   4;
%e A350824   0, 1,   8;
%e A350824   0, 1,  20,   36;
%e A350824   0, 1,  24,   36;
%e A350824   0, 1,  36,   72;
%e A350824   0, 1,  52,  108;
%e A350824   0, 1, 112,  576,  576;
%e A350824   0, 1, 128,  612,  576;
%e A350824   0, 1, 200, 1116, 1152;
%e A350824   ...
%e A350824 The T(5,1) = 1 pattern is 11111.
%e A350824 The T(5,2) = 8 patterns are 12222, 11222, 11122, 11112, 21111, 22111, 22211, 22221.
%o A350824 (PARI)
%o A350824 P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
%o A350824 R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
%o A350824 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]}
%o A350824 { my(A=T(16)); for(n=1, #A, print(A[n])) }
%Y A350824 Row sums are A351292.
%Y A350824 Cf. A000217, A001044, A003056, A008289, A351637, A351640.
%K A350824 nonn,tabf
%O A350824 0,8
%A A350824 _Andrew Howroyd_, Feb 12 2022