A084851 Binomial transform of binomial(n+2,2).

1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, 22784, 52736, 120832, 274432, 618496, 1384448, 3080192, 6815744, 15007744, 32899072, 71827456, 156237824, 338690048, 731906048, 1577058304, 3388997632, 7264534528, 15535702016, 33151778816

Offset

0,2

Comments

Essentially the same as A049611.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Igor Makhlin, Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties, arXiv:2003.02916 [math.CO], 2020.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Formula

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

a(n) = (n^2 + 7*n + 8)*2^(n - 3).

a(n) = Sum_{k=0..n} C(n, k)*C(k+2, 2).

a(n) = A049611(n+1).

Example

From Bruno Berselli, Jul 17 2018: (Start)
Let the triangle:
   1
   3,  4
   6,  9,  13
  10, 16,  25,  38
  15, 25,  41,  66, 104
  21, 36,  61, 102, 168, 272
  28, 49,  85, 146, 248, 416,  688
  36, 64, 113, 198, 344, 592, 1008, 1696, etc.
where the first column is A000217 (without 0). The other terms are calculated with the recurrence T(r, c) = T(r-1, c-1) + T(r, c-1).
The sequence is the right side of the triangle.
(End)

Maple

a := n -> hypergeom([-n, 3], [1], -1);
seq(round(evalf(a(n), 32)), n=0..31); # Peter Luschny, Aug 02 2014

Mathematica

CoefficientList[ Series[(1 - x)^2/(1 - 2 x)^3, {x, 0, 28}], x] (* Robert G. Wilson v, Jun 28 2005 *)
LinearRecurrence[{6, -12, 8}, {1, 4, 13}, 30] (* Harvey P. Dale, Aug 05 2019 *)

Prog

(Magma) [(n^2+7*n+8)*2^(n-3): n in [0..40]]; // Vincenzo Librandi, Aug 03 2014

Crossrefs

Cf. A000217, A049611, A058396 (first differences).

Sequence in context: A089092 A181527 A049611 * A094706 A325927 A056014

Adjacent sequences: A084848 A084849 A084850 * A084852 A084853 A084854

Keyword

nonn,easy

Author

Paul Barry, Jun 09 2003

Status

approved