A085458 -id:A085458 - cp's OEIS frontend

A085456 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-7)^n.

1, -4, 20, -116, 708, -4452, 28532, -185300, 1215268, -8030404, 53381844, -356577588, 2391430020, -16092704292, 108605848116, -734783381652, 4982063186916, -33844621986180, 230306722637204, -1569571734301172, 10711405584991300, -73188920628617956, 500643475619050740

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jul 01 2003

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A085455, A085457, A085458.

CoefficientList[Series[Sqrt[(1-x)/(1+7 x)], {x, 0, 30}], x]

a(n) = sum(k=0, n, (-2)^k*binomial(n-1, n-k)*binomial(2*k, k)); \\ Seiichi Manyama, Feb 03 2023

G.f.: A(x)=Sqrt((1-x)/(1+7x)).
From Seiichi Manyama, Feb 03 2023: (Start)
a(n) = Sum_{k=0..n} (-2)^k * binomial(n-1,n-k) * binomial(2*k,k).
n*a(n) = -2*(3*n-1)*a(n-1) + 7*(n-2)*a(n-2). (End)

A383947 Expansion of 1/sqrt((1+x)^3 * (1-7*x)).

Original entry on oeis.org

1, 2, 15, 84, 525, 3318, 21371, 139240, 915417, 6060330, 40345767, 269825724, 1811432805, 12200012958, 82394389395, 557794589904, 3784079617713, 25718668160850, 175085306697791, 1193682452744740, 8148955372804029, 55697327430265158, 381099865385716395

Offset: 0

Seiichi Manyama, Aug 19 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A085458, A322242.

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1+x)^3 * (1-7*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

CoefficientList[Series[1/Sqrt[(1+x)^3*(1-7*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1+x)^3*(1-7*x)))

n*a(n) = (6*n-4)*a(n-1) + 7*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-2)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(2*k,k) * binomial(n+1,n-k).

cp's OEIS Frontend

A085456 Sum_{i=0..n} Sum_{j=0..i} a(j) * a(i-j) = (-7)^n.

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