A165189 Partial sums of partial sums of (A001840 interleaved with zeros).

1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132, 171, 210, 264, 318, 390, 462, 556, 650, 770, 890, 1040, 1190, 1375, 1560, 1785, 2010, 2280, 2550, 2871, 3192, 3570, 3948, 4389, 4830, 5341, 5852, 6440, 7028, 7700, 8372, 9136, 9900, 10764, 11628, 12600, 13572

Offset

1,2

Comments

Also convolution of period six sequence 1,0,0,0,0,0,1,... (A079979) with sequence 1,2,5,8,14,20,30,40,... (A006918 without initial zero).

Links

Table of n, a(n) for n=1..48.

Index entries for linear recurrences with constant coefficients, signature (2, 1, -4, 1, 2, 0, -2, -1, 4, -1, -2, 1).

Formula

G.f.: x/((1-x)^5*(1+x)^3*(1-x+x^2)*(1+x+x^2)).

54*a(n) = 631/64 +405/16*n +3/32*n^4 +15/8*n^3 +381/32*n^2 -(-1)^n*( 9/32*n^2 +45/16*n +375/64) -A131713(n) -3*A057079(n). - R. J. Mathar, Jun 16 2018

Example

A001840 interleaved with zeros is
1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 9, 0, 12, 0, 15, 0, ...
Partial sums thereof are
1, 1, 3, 3, 6, 6, 11, 11, 18, 18, 27, 27, 39, 39, 54, 54, ...
This equals A014125 interleaved with itself.
Partial sums thereof are
1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132, 171, 210, 264, 318, ...

Mathematica

Drop[Accumulate[Accumulate[Riffle[LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 2, 3, 5}, 30], 0]]], 2] (* or *) LinearRecurrence[{2, 1, -4, 1, 2, 0, -2, -1, 4, -1, -2, 1}, {1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132}, 50] (* Harvey P. Dale, Jun 08 2018 *)

Prog

(PARI) /* first computes u = A001840 interleaved with zeros, then v = partial sums, then w = second partial sums */ {m=50; u=vector(m, n, polcoeff(x/((1-x^2)^3*(1+x^2+x^4))+x*O(x^(n)), n)); v=vector(m); a=u[1]; v[1]=a; for(n=2, m, a+=u[n]; v[n]=a); w=vector(m-1); a=v[1]; w[1]=a; for(n=2, m-1, a+=v[n]; w[n]=a); w} \\ Klaus Brockhaus, Sep 21 2009

Crossrefs

Cf. A001840 (expansion of x/((1-x)^3*(1+x+x^2))), A001840 (expansion of x/((1-x)^2*(1-x^3))), A079979, A006918, A014125.

Sequence in context: A022907 A006918 A274523 * A358055 A011842 A000094

Adjacent sequences: A165186 A165187 A165188 * A165190 A165191 A165192

Keyword

nonn

Author

Alford Arnold, Sep 16 2009

Extensions

Edited and corrected by R. J. Mathar, Klaus Brockhaus and N. J. A. Sloane, Sep 21 2009 - Sep 25 2009

Status

approved