A239632 Number of parts in all palindromic compositions of n.

0, 1, 3, 4, 10, 12, 28, 32, 72, 80, 176, 192, 416, 448, 960, 1024, 2176, 2304, 4864, 5120, 10752, 11264, 23552, 24576, 51200, 53248, 110592, 114688, 237568, 245760, 507904, 524288, 1081344, 1114112, 2293760, 2359296, 4849664, 4980736, 10223616, 10485760, 21495808, 22020096, 45088768, 46137344

Offset

0,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Formula

G.f.: (x + 3*x^2 - 2*x^4)/(1 - 2*x^2)^2.

a(n) = Sum_{k=1..n} A051159(n,k)*k.

a(n) = 4*a(n-2) - 4*a(n-4) for n > 3. - Giovanni Resta, Mar 23 2014

a(2k) = (2k+1)*2^(k-1) for k>0, a(2k+1) = (2k+2)*2^(k-1) for k>=0. - Gregory L. Simay, Dec 05 2022

E.g.f.: (2*(1 + x)*cosh(sqrt(2)*x) + sqrt(2)*(1 + 2*x)*sinh(sqrt(2)*x) - 2)/4. - Stefano Spezia, Apr 25 2024

Example

a(5)=12 because we have: 5, 1+3+1, 2+1+2, 1+1+1+1+1 with a total of 12 parts.

Mathematica

nn=30; r=Solve[p==y/(1-x) - y + 1 + y^2*x^2/(1-x^2)*p, p]; CoefficientList[Series[D[p/.r, y]/.y->1, {x, 0, nn}], x]
CoefficientList[Series[(x + 3 x^2 - 2 x^4)/(1 - 2 x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 23 2014 *)

Crossrefs

Cf. A051159.

Sequence in context: A362553 A360620 A092434 * A031367 A073443 A257494

Adjacent sequences: A239629 A239630 A239631 * A239633 A239634 A239635

Keyword

nonn,easy

Author

Geoffrey Critzer, Mar 22 2014

Status

approved