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G.f.: 1/sqrt( (1 + 3*x)*(1 - 13*x) ).
4

%I #22 Jan 21 2025 09:24:06

%S 1,5,57,605,6961,81525,973545,11765325,143522145,1763351525,

%T 21789466777,270509191485,3371353189009,42155188480085,

%U 528587607974217,6644129071092525,83691484792766145,1056178325362832325,13351036742005533945,169019946403985898525,2142600388730167543281,27193744661180635582005,345520219114720175821737,4394534009569783690837005,55943630366450131877449761,712778930909503993783945125

%N G.f.: 1/sqrt( (1 + 3*x)*(1 - 13*x) ).

%H Seiichi Manyama, <a href="/A322248/b322248.txt">Table of n, a(n) for n = 0..899</a>

%H Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

%F a(n) = Sum_{k=0..n} 13^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).

%F a(n) = Sum_{k=0..n} (-3)^(n-k) * 4^k * binomial(n,k)*binomial(2*k,k).

%F a(n) equals the (central) coefficient of x^n in (1 + 5*x + 16*x^2)^n.

%F a(n) ~ 13^(n + 1/2) / (4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Dec 10 2018

%F D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) + 39*(n-1)*a(n-2) for n > 1. - _Seiichi Manyama_, Apr 22 2019

%e G.f.: A(x) = 1 + 5*x + 57*x^2 + 605*x^3 + 6961*x^4 + 81525*x^5 + 973545*x^6 + 11765325*x^7 + 143522145*x^8 + 1763351525*x^9 + 21789466777*x^10 + ...

%e such that A(x)^2 = 1/(1 - 10*x - 39*x^2).

%e RELATED SERIES.

%e exp( Sum_{n>=1} a(n)*x^n/n ) = 1 + 5*x + 41*x^2 + 365*x^3 + 3537*x^4 + 35925*x^5 + 378105*x^6 + 4084925*x^7 + 45044129*x^8 + 504880805*x^9 + 5735247817*x^10 + ...

%t CoefficientList[Series[1/Sqrt[((1+3x)(1-13x))],{x,0,30}],x] (* _Harvey P. Dale_, Jun 29 2021 *)

%o (PARI) /* Using generating function: */

%o {a(n) = polcoeff( 1/sqrt((1 + 3*x)*(1 - 13*x) +x*O(x^n)),n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) /* Using binomial formula: */

%o {a(n) = sum(k=0,n, (-3)^(n-k)*4^k*binomial(n,k)*binomial(2*k,k))}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) /* Using binomial formula: */

%o {a(n) = sum(k=0,n, 13^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) /* a(n) as a central coefficient */

%o {a(n) = polcoeff( (1 + 5*x + 16*x^2 +x*O(x^n))^n, n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A322249.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 10 2018