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A055270 a(n) = 7*a(n-1) + (-1)^n * binomial(2,2-n) with a(-1)=0.

%I A055270 #36 Sep 08 2022 08:45:01
%S A055270 1,5,36,252,1764,12348,86436,605052,4235364,29647548,207532836,
%T A055270 1452729852,10169108964,71183762748,498286339236,3488004374652,
%U A055270 24416030622564,170912214357948,1196385500505636,8374698503539452,58622889524776164,410360226673433148,2872521586714032036
%N A055270 a(n) = 7*a(n-1) + (-1)^n * binomial(2,2-n) with a(-1)=0.
%C A055270 For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7} we have f(x_1) <> y_1 and f(x_2) <> y_2. - _Milan Janjic_, Apr 19 2007
%C A055270 a(n) is the number of generalized compositions of n when there are 6*i-1 different types of i, (i=1,2,...). - _Milan Janjic_, Aug 26 2010
%D A055270 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H A055270 G. C. Greubel, <a href="/A055270/b055270.txt">Table of n, a(n) for n = 0..1000</a>
%H A055270 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A055270 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (7).
%F A055270 a(n) = 6^2 * 7^(n-2), n >= 2 with a(0)=1, a(1)=5.
%F A055270 G.f.: (1-x)^2/(1-7*x).
%F A055270 a(n) = Sum_{k=0..n} A201780(n,k)*5^k. - _Philippe Deléham_, Dec 05 2011
%F A055270 E.g.f.: (13 - 7*x + 36*exp(7*x))/49. - _G. C. Greubel_, Mar 16 2020
%p A055270 A055270:= n-> `if`(n<2, 4*n+1, 36*7^(n-2)); seq(A055270(n), n=0..30); # _G. C. Greubel_, Mar 16 2020
%t A055270 Join[{1,5},NestList[7#&,36,20]] (* _Harvey P. Dale_, Sep 04 2017 *)
%o A055270 (Magma) [1,5] cat [36*7^(n-2): n in [2..30]]; // _G. C. Greubel_, Mar 16 2020
%o A055270 (Sage) [1,5]+[36*7^(n-2) for n in (2..30)] # _G. C. Greubel_, Mar 16 2020
%Y A055270 Cf. A055272 (first differences of 7^n (A000420)).
%K A055270 easy,nonn
%O A055270 0,2
%A A055270 _Barry E. Williams_, May 10 2000
%E A055270 Terms a(20) onward added by _G. C. Greubel_, Mar 16 2020