cp's OEIS Frontend

A355129 a(n) is the number of integer sequences b(0..n) of length n+1, with 0 <= b(k) <= k! and monotonic b(k) <= b(k+1).

%I A355129 #65 Aug 06 2022 08:05:36
%S A355129 2,3,7,40,856,91821,60080136,279276911843,10503211888973754,
%T A355129 3585680755683196123365,12323227994417456429490342865,
%U A355129 468378989392773003347310901356953089,214565221409985003242070442557341938941878313,1282499669290042152350268651085002913530161723080398635
%N A355129 a(n) is the number of integer sequences b(0..n) of length n+1, with 0 <= b(k) <= k! and monotonic b(k) <= b(k+1).
%C A355129 List of the possible cases regarding the patterns of the numbers in the sequence b:
%C A355129 Length:  1    2    3    4    5    6
%C A355129 Pos 0:   1    1    1    1    1    1
%C A355129 Pos 1:   1    2    3    4    5    6
%C A355129 Pos 2:   0    0    3    7   12   18
%C A355129 Pos 3:   0    0    0    7   19   37
%C A355129 Pos 4:   0    0    0    7   26   63
%C A355129 Pos 5:   0    0    0    7   33   96
%C A355129 Pos 6:   0    0    0    7   40  136
%C A355129 Pos 7:   0    0    0    0   40  176
%C A355129 Pos 8:   0    0    0    0   40  216
%C A355129    ...  ...  ...  ...  ... ...  ...
%C A355129     Sum: 2    3    7   40  856 91821
%C A355129 Each row counts the number of possible distributions of numbers, row "Pos 0" is the number of possible distributions with only the number zero. The row "Pos 1" counts the distributions of zeros and ones. The row "Pos 2" the possible distributions of {0,1,2} and so forth.
%C A355129 From top to down: If a number in the column length = k has reached the value of the sum of the column length = k-1, this number will be k!-(k-1)!+1 times repeated. Before this limit is reached each number is the sum of the neighbor one step above and the neighbor one step to the left.
%F A355129 a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + Sum_{r = 1..n-2} Sum_{k = 0..r-1} binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1).
%e A355129 For a(0) we get two possible sequences:
%e A355129   {0}, {1}.
%e A355129 For a(1) we get three possible sequences:
%e A355129   {0, 0}, {0, 1}, {1, 1}.
%e A355129 For a(2) = 7 we get:
%e A355129   {0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 1, 1},
%e A355129   {0, 1, 2}, {1, 1, 1}, {1, 1, 2}.
%o A355129 (PARI)
%o A355129 a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + sum(r = 1, n-2, sum(k = 0, r-1 ,binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1)))
%Y A355129 Cf. A000108 (if we change the definition into 0 <= b(k) <= k).
%Y A355129 Cf. A005269, A005270, A008934, A016121, A128094, A242105.
%K A355129 nonn
%O A355129 0,1
%A A355129 _Thomas Scheuerle_, Aug 04 2022