A335648 Partial sums of A006010.

0, 1, 6, 26, 78, 195, 420, 820, 1476, 2501, 4026, 6222, 9282, 13447, 18984, 26216, 35496, 47241, 61902, 80002, 102102, 128843, 160908, 199068, 244140, 297037, 358722, 430262, 512778, 607503, 715728, 838864, 978384, 1135889, 1313046, 1511658, 1733598, 1980883, 2255604

Offset: 0

Author

Stefano Spezia, Jun 15 2020

Keywords

• nonn
• easy

Links

• Stefano Spezia, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,10,-4,-4,4,-1).
• Index entries for partial sums.

Crossrefs

Cf. A000290, A001844, A005408, A007204, A053755, A058331, A059722.

Cf. A006010 (1st differences), A186424 (3rd differences), A317614 (2nd differences).

Programs

Magma

I:=[0, 1, 6, 26, 78, 195, 420, 820]; [n le 8 select I[n] else 4*Self(n-1)-4*Self(n-2)-4*Self(n-3)+10*Self(n-4)-4*Self(n-5)-4*Self(n-6)+4*Self(n-7)-Self(n-8): n in [1..39]];

Mathematica

Table[(1+n)(5-5(-1)^n+8n+12n^2+8n^3+2n^4)/80,{n,0,38}]

PARI

a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80;

Sage

(x*(1+2*x+6*x^2+2*x^3+x^4)/((1-x)^6*(1+x)^2)).series(x, 39).coefficients(x, False)

Formula

a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80.

O.g.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^6*(1 + x)^2).

E.g.f.: (cosh(x) - sinh(x))*(-5 + 5*x + (5 + 65*x + 180*x^2 + 130*x^3 + 30*x^4 + 2*x^5)*(cosh(2*x) + sinh(2*x)))/80.

a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n > 7.

a(2*n-1) = n*A053755(n)/5 for n > 0.

a(2*n) = n*A005408(n)*A059722(n-1)/5.

a(2*n+1) - a(2*n-1) = A001844(n)^2 = A007204(n) for n > 0.

a(2*n) - a(2*n-2) = 2*A000290(n)*A058331(n) for n > 0.