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A112532
First differences of [0, A047970].
5
1, 1, 3, 9, 29, 101, 379, 1525, 6549, 29889, 144419, 736241, 3947725, 22201549, 130624587, 802180701, 5131183301, 34121977865, 235486915507, 1683925343929, 12458499203901, 95237603403381, 751291094637083, 6108883628141189, 51144808472958709, 440444879385258001
OFFSET
0,3
COMMENTS
Number of sequences of length n in [n] (endofunctions) whose first run has length equal to the maximum of the sequence.
LINKS
FORMULA
G.f.: (1-x)^2*( Sum_{n >= 0} x^n/(1 - (n+2)*x) ). - Peter Bala, Jul 09 2014
From Mathew Englander, Feb 28 2021: (Start)
a(n) = A089246(n+2,0) - A089246(n+1,0).
a(n) = n + Sum_{i = 0..n} (n-i-1)^2 * (n-i)^i. (End)
EXAMPLE
The 9 sequences for n=4 (sorted by maximum)
1121,1122,2211,2212, 1113,2223,3331,3332, 4444
The 29 sequences for n=5 (sorted by maximum)
11211,11212,11221,11222, 22111,22112,22121,22122, 11123,11131,11132,11133, 22213,22231,22232,22233, 33311,33312,33313,33321,33322,33323, 11114, 22224, 33334, 44441,44442,44443, 55555
MATHEMATICA
a[n_]:= If[n==0, 1, n + Sum[(i-1)^2*i^(n-i), {i, 0, n}]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 12 2022 *)
PROG
(PARI) a(n) = n + sum(i = 0, n, (n-i-1)^2 * (n-i)^i); \\ Michel Marcus, Mar 01 2021
(Sage) [n +sum((j-1)^2*j^(n-j) for j in (0..n)) for n in (0..30)] # G. C. Greubel, Jan 12 2022
CROSSREFS
First differences of column 0 of triangle A089246 (beginning at row 1). With offset 1, first differences of column 0 of triangle A242431. Second differences of column 0 of triangle A101494.
Sequence in context: A278404 A148941 A079319 * A148942 A109432 A148943
KEYWORD
nonn
AUTHOR
Alford Arnold, Sep 10 2005
EXTENSIONS
Corrected by D. S. McNeil, Aug 20 2010
Combinatorial interpretation and examples by Olivier Gérard, Jan 29 2023
STATUS
approved