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%I #12 Jan 03 2021 06:40:18
%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);
%t b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, 1, If[i == 0, Sum[b[n - h, j, h, k], {h, 1, n}], Sum[b[n - h, j, h, k], {h, Max[1, 2*j - i - k], Min[n, 2*j - i + k]}]]];
%t a[n_] := If[n == 0, 1, b[n - 1, 0, 1, n]];
%t a /@ Range[0, 40] (* _Jean-François Alcover_, Jan 03 2021, after _Alois P. Heinz_ *)
%Y Main diagonal of A239550.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Mar 21 2014
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