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 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). 0
 2, 3, 7, 40, 856, 91821, 60080136, 279276911843, 10503211888973754, 3585680755683196123365, 12323227994417456429490342865, 468378989392773003347310901356953089, 214565221409985003242070442557341938941878313, 1282499669290042152350268651085002913530161723080398635 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS List of the possible cases regarding the patterns of the numbers in the sequence b: Length:  1    2    3    4    5    6 Pos 0:   1    1    1    1    1    1 Pos 1:   1    2    3    4    5    6 Pos 2:   0    0    3    7   12   18 Pos 3:   0    0    0    7   19   37 Pos 4:   0    0    0    7   26   63 Pos 5:   0    0    0    7   33   96 Pos 6:   0    0    0    7   40  136 Pos 7:   0    0    0    0   40  176 Pos 8:   0    0    0    0   40  216    ...  ...  ...  ...  ... ...  ...     Sum: 2    3    7   40  856 91821 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. 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. LINKS FORMULA 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). EXAMPLE For a(0) we get two possible sequences:   {0}, {1}. For a(1) we get three possible sequences:   {0, 0}, {0, 1}, {1, 1}. For a(2) = 7 we get:   {0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 1, 1},   {0, 1, 2}, {1, 1, 1}, {1, 1, 2}. PROG (PARI) 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))) CROSSREFS Cf. A000108 (if we change the definition into 0 <= b(k) <= k). Cf. A005269, A005270, A008934, A016121, A128094, A242105. Sequence in context: A182219 A227777 A037843 * A102604 A119662 A265776 Adjacent sequences:  A355126 A355127 A355128 * A355130 A355131 A355132 KEYWORD nonn AUTHOR Thomas Scheuerle, Aug 04 2022 STATUS approved

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Last modified September 29 22:48 EDT 2022. Contains 357092 sequences. (Running on oeis4.)