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

0,2

Comments

The sequence is column 3 of triangle in A207630.

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

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).

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).

Example

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A212339 A072111 A173627 * A141397 A305779 A143131

Adjacent sequences: A295285 A295286 A295287 * A295289 A295290 A295291

Keyword

nonn,easy

Author

Franck Maminirina Ramaharo, Nov 19 2017

Status

approved