login
A335648
Partial sums of A006010.
2
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
0,3
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.
MATHEMATICA
Table[(1+n)(5-5(-1)^n+8n+12n^2+8n^3+2n^4)/80, {n, 0, 38}]
PROG
(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]];
(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)
CROSSREFS
Cf. A006010 (1st differences), A186424 (3rd differences), A317614 (2nd differences).
Sequence in context: A175898 A255870 A286188 * A094162 A229572 A316160
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Jun 15 2020
STATUS
approved