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

0,3

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.

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

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

Sequence in context: A175898 A255870 A286188 * A094162 A229572 A316160

Adjacent sequences: A335645 A335646 A335647 * A335649 A335650 A335651

Keyword

nonn,easy

Author

Stefano Spezia, Jun 15 2020

Status

approved