A322242 G.f.: 1/sqrt(1 - 6*x - 7*x^2).

1, 3, 17, 99, 609, 3843, 24689, 160611, 1054657, 6975747, 46406097, 310171491, 2081258529, 14011445763, 94594402353, 640188979299, 4341874207617, 29502747778563, 200803974858641, 1368767759442531, 9342637825548769, 63846282803069187, 436797192815981553, 2991302112253485411, 20504081077963103041, 140665546932766467843, 965770879590646638929, 6635507385062085656931, 45621050527781298148257

Offset: 0

Author

Paul D. Hanna, Dec 08 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 3*x + 17*x^2 + 99*x^3 + 609*x^4 + 3843*x^5 + 24689*x^6 + 160611*x^7 + 1054657*x^8 + 6975747*x^9 + 46406097*x^10 + ...
such that A(x)^2 = 1/(1 - 6*x - 7*x^2).
RELATED SERIES.
Ignoring the initial term, this sequence yields the logarithmic derivative of
exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 3*x + 13*x^2 + 63*x^3 + 329*x^4 + 1803*x^5 + 10229*x^6 + 59559*x^7 + 353873*x^8 + 2136915*x^9 + 13076637*x^10 + ...
which equals (1-3*x - sqrt(1 - 6*x - 7*x^2))/(8*x^2).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1184
• Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Crossrefs

Cf. A322243.

Programs

Maple

f:= gfun:-rectoproc({(7*n+7)*a(n)+(9+6*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 3}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Dec 10 2018

Mathematica

CoefficientList[Series[1/Sqrt[1-6x-7x^2],{x,0,40}],x] (* Harvey P. Dale, Apr 14 2019 *)

PARI

/* Using generating function: */
{a(n) = polcoeff( 1/sqrt(1 - 6*x - 7*x^2 +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

PARI

/* Using binomial formula: */
{a(n) = sum(k=0,n, (-1)^(n-k)*2^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

PARI

/* Using binomial formula: */
{a(n) = sum(k=0,n, 7^(n-k)*(-2)^k*binomial(n,k)*binomial(2*k,k))}
for(n=0,30,print1(a(n),", "))

PARI

/* a(n) is central coefficient in (1 + 3*x + 4*x^2)^n */
{a(n) = polcoeff( (1 + 3*x + 4*x^2 +x*O(x^n))^n, n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k).

a(n) = Sum_{k=0..n} 7^(n-k) * (-2)^k * binomial(n,k)*binomial(2*k,k).

a(n) equals the (central) coefficient of x^n in (1 + 3*x + 4*x^2)^n.

exp( Sum_{n>=1} a(n)*x^n/n ) = (1-3*x - sqrt(1 - 6*x - 7*x^2))/(8*x^2).

D-finite with recurrence: (7*n+7)*a(n) + (9+6*n)*a(n+1) + (-n-2)*a(n+2) = 0. - Robert Israel, Dec 10 2018

a(n)^2 = A322243(n), which gives the coefficients in 1 / AGM(1+7*x, sqrt((1-x)*(1-7^2*x))). - Paul D. Hanna, Apr 20 2019

a(n) ~ 7^(n + 1/2) / (2^(3/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 27 2019

E.g.f.: exp(3*x) * BesselI(0,4*x). - Ilya Gutkovskiy, Feb 02 2021

a(n) = 7^n*2F1([1/2, -n], [1], 8/7), where 2F1 is the hypergeometric function. - Stefano Spezia, Feb 02 2021

a(n) = (1/4)^n * Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025