%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^(n2).  _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*n348) *b(n1)
%p +(2*n^4+23*n^3155*n^2166*n+3776) *b(n2)
%p +(n^4+14*n^35*n^2+122*n+768) *b(n3)
%p +(2*n^3+10*n^264*n1328) *b(n4)
%p (2*n^4+28*n^378*n^2272*n+2320) *b(n5))/
%p (n^4+10*n^375*n^220*n+1244))
%p end:
%p a:= n> `if`(n<7, ceil(2^(n2)), 2^(n2)b(n7)):
%p seq(a(n), n=0..40);
%Y Main diagonal of A239550.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Mar 21 2014
