A295288 Binomial transform of the centered triangular numbers A005448.

1, 5, 19, 62, 184, 512, 1360, 3488, 8704, 21248, 50944, 120320, 280576, 647168, 1478656, 3350528, 7536640, 16842752, 37421056, 82706432, 181927936, 398458880, 869269504, 1889533952, 4093640704, 8841592832, 19042140160, 40902852608

Offset: 0

Author

Franck Maminirina Ramaharo, Nov 19 2017

Keywords

• nonn
• easy

Comments

The sequence is column 3 of triangle in A207630.

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

Examples

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

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

Programs

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

Maple

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

Mathematica

Table[(3 n^2 + 9 n + 8) 2^(n-3), {n, 0, 70}]
LinearRecurrence[{6,-12,8}, {1,5,19}, 50] (* G. C. Greubel, Oct 17 2018 *)

Maxima

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

PARI

a(n) = (3*n^2 + 9*n + 8)*2^(n - 3) \\ Felix Fröhlich, Nov 19 2017

Formula

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

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

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

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

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

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