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A239561 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-n,...,n}. 2

%I

%S 1,1,1,2,4,8,16,31,63,125,252,504,1013,2027,4069,8141,16318,32650,

%T 65381,130801,261791,523677,1047780,2095796,4192533,8385623,16773321,

%U 33547917,67100362,134203614,268417029,536840509,1073702131,2147418493,4294882224,8589795592

%N Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-n,...,n}.

%H Alois P. Heinz, <a href="/A239561/b239561.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ 2^(n-2). - _Vaclav Kotesovec_, May 01 2014

%e There are 2^5 = 32 compositions of 7 with first part = 1. Exactly one of these has second differences not in {-7,...,7}, namely [1,5,1]. Thus a(7) = 32 - 1 = 31.

%p b:= proc(n) option remember; `if`(n<5, [1, 1, 3, 4, 8][n+1],

%p (-(n^3+3*n^2+184*n-348) *b(n-1)

%p +(2*n^4+23*n^3-155*n^2-166*n+3776) *b(n-2)

%p +(n^4+14*n^3-5*n^2+122*n+768) *b(n-3)

%p +(2*n^3+10*n^2-64*n-1328) *b(n-4)

%p -(2*n^4+28*n^3-78*n^2-272*n+2320) *b(n-5))/

%p (n^4+10*n^3-75*n^2-20*n+1244))

%p end:

%p a:= n-> `if`(n<7, ceil(2^(n-2)), 2^(n-2)-b(n-7)):

%p seq(a(n), n=0..40);

%Y Main diagonal of A239550.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Mar 21 2014

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Last modified April 9 20:46 EDT 2020. Contains 333363 sequences. (Running on oeis4.)