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

Author

Paul Barry, Jun 09 2003

Keywords

• nonn
• easy

Comments

Essentially the same as A049611.

Examples

			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)
		

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).

Crossrefs

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

Programs

Magma

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

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 *)

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).