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A295288 Binomial transform of the centered triangular numbers A005448. 2

%I

%S 1,5,19,62,184,512,1360,3488,8704,21248,50944,120320,280576,647168,

%T 1478656,3350528,7536640,16842752,37421056,82706432,181927936,

%U 398458880,869269504,1889533952,4093640704,8841592832,19042140160,40902852608

%N Binomial transform of the centered triangular numbers A005448.

%C The sequence is column 3 of triangle in A207630.

%C First difference is given by A055818(n+3,3) for n > 0.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

%H G. C. Greubel, <a href="/A295288/b295288.txt">Table of n, a(n) for n = 0..1000</a>

%H Luis Manuel Rivera, <a href="https://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv:1406.3081 [math.CO], 2014.

%H C. Corsani, D. Merlini, R. Sprugnoli, <a href="https://doi.org/10.1016/S0012-365X(97)00110-6">Left-inversion of combinatorial sums</a>, Discrete Mathematics, 180 (1998) 107-122.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).

%F G.f.: (1 - x + x^2)/(1 - 2*x)^3.

%F a(n+3) = 8*a(n) - 12*a(n+1) + 6*a(n+2).

%F a(n+1) = 2*a(n) + 3*(n + 2)*2^(n-1).

%F a(n+1) = 2*a(n) + 3*A001792(n) = 2*a(n) + A001787(n+2) - A001792(n).

%F a(n) = (3*n^2 + 9*n + 8)*2^(n - 3).

%F a(n) = (1/8)*A077588(n+2)*A000079(n).

%e a(0) = (3*0^2 + 9*0 + 8)*2^(-3) = 8/8 = 1.

%p A:=n->(3*n^2+9*n+8)*2^(n-3); seq(A(n), n=0..70);

%t Table[(3 n^2 + 9 n + 8) 2^(n-3), {n, 0, 70}]

%t LinearRecurrence[{6,-12,8}, {1,5,19}, 50] (* _G. C. Greubel_, Oct 17 2018 *)

%o (Maxima) makelist((3*n^2 + 9*n + 8)*2^(n - 3), n, 0, 70);

%o (PARI) a(n) = (3*n^2 + 9*n + 8)*2^(n - 3) \\ _Felix Fröhlich_, Nov 19 2017

%o (MAGMA) I:=[1,5,19]; [n le 3 select I[n] else 6*Self(n-1) -12*Self(n-2) +8*Self(n-3): n in [1..40]]; // _G. C. Greubel_, Oct 17 2018

%Y Cf. A000079, A001787, A001792, A005448, A055818, A077588, A207629, A207630.

%K nonn,easy

%O 0,2

%A _Franck Maminirina Ramaharo_, Nov 19 2017

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Last modified June 26 20:30 EDT 2019. Contains 324380 sequences. (Running on oeis4.)