This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A165189 #14 Jun 16 2018 12:35:46
%S A165189 1,2,5,8,14,20,31,42,60,78,105,132,171,210,264,318,390,462,556,650,
%T A165189 770,890,1040,1190,1375,1560,1785,2010,2280,2550,2871,3192,3570,3948,
%U A165189 4389,4830,5341,5852,6440,7028,7700,8372,9136,9900,10764,11628,12600,13572
%N A165189 Partial sums of partial sums of (A001840 interleaved with zeros).
%C A165189 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).
%H A165189 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -4, 1, 2, 0, -2, -1, 4, -1, -2, 1).
%F A165189 G.f.: x/((1-x)^5*(1+x)^3*(1-x+x^2)*(1+x+x^2)).
%F A165189 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
%e A165189 A001840 interleaved with zeros is
%e A165189 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 9, 0, 12, 0, 15, 0, ...
%e A165189 Partial sums thereof are
%e A165189 1, 1, 3, 3, 6, 6, 11, 11, 18, 18, 27, 27, 39, 39, 54, 54, ...
%e A165189 This equals A014125 interleaved with itself.
%e A165189 Partial sums thereof are
%e A165189 1, 2, 5, 8, 14, 20, 31, 42, 60, 78, 105, 132, 171, 210, 264, 318, ...
%t A165189 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 *)
%o A165189 (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
%Y A165189 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.
%K A165189 nonn
%O A165189 1,2
%A A165189 _Alford Arnold_, Sep 16 2009
%E A165189 Edited and corrected by _R. J. Mathar_, _Klaus Brockhaus_ and _N. J. A. Sloane_, Sep 21 2009 - Sep 25 2009