A143131 Binomial transform of [1, 4, 10, 20, 0, 0, 0, ...].

1, 5, 19, 63, 157, 321, 575, 939, 1433, 2077, 2891, 3895, 5109, 6553, 8247, 10211, 12465, 15029, 17923, 21167, 24781, 28785, 33199, 38043, 43337, 49101, 55355, 62119, 69413, 77257, 85671, 94675, 104289, 114533, 125427, 136991, 149245

Offset

1,2

Links

Table of n, a(n) for n=1..37.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

Binomial transform of [1, 4, 10, 20, 0, 0, 0, ...], where (1, 4, 10, 20) = row 3 of triangle A046899

a(n) = (-39 + 77*n - 45*n^2 + 10*n^3)/3. - T. D. Noe, Aug 22 2008

G.f.: x*(1+x+5*x^2+13*x^3)/(1-x)^4. - Colin Barker, Mar 23 2012

Example

a(4) = 63 = (1, 3, 3, 1) dot (1, 4, 10, 20) = (1 + 12 + 30 + 20).

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 5, 19, 63}, 37] (* or *) Rest[CoefficientList[Series[x*(1+x+5*x^2+13*x^3)/(1-x)^4, {x, 0, 37}], x]] (* or *) a[n_]:=(-39 + 77*n - 45*n^2 + 10*n^3)/3; Array[a, 37] (* James C. McMahon, Aug 17 2025 *)

Crossrefs

Cf. A046899.

Sequence in context: A295288 A141397 A305779 * A036677 A003296 A053545

Adjacent sequences: A143128 A143129 A143130 * A143132 A143133 A143134

Keyword

nonn,easy

Author

Gary W. Adamson, Jul 27 2008

Extensions

More terms from T. D. Noe, Aug 22 2008

Status

approved