A387367 - OEIS

%I #24 Oct 04 2025 16:18:53

%S 1,25,435,6475,88270,1137150,14081970,169370850,1991916135,

%T 23013193775,262062237437,2948690451525,32845189782860,

%U 362721036253100,3975956599494420,43300257350934900,468875116313950845,5051523021827188725,54177811767428268535,578700942412768257775

%N Expansion of 1/(1 - 10*x + x^2)^(5/2).

%H Vincenzo Librandi, <a href="/A387367/b387367.txt">Table of n, a(n) for n = 0..1000</a>

%F n*a(n) = 5*(2*n+3)*a(n-1) - (n+3)*a(n-2) for n > 1.

%F a(n) = (binomial(n+4,2)/6) * A387369(n).

%F a(n) = (-1)^n * Sum_{k=0..n} (1/10)^(n-2*k) * binomial(-5/2,k) * binomial(k,n-k).

%t CoefficientList[Series[1/(1-10*x+x^2)^(5/2),{x,0,33}],x] (* _Vincenzo Librandi_, Aug 29 2025 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(1/(1-10*x+x^2)^(5/2))

%o (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 10*x + x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 29 2025

%o (Python)

%o from sympy import Symbol, series, Rational

%o x = Symbol('x')

%o def A387367(n): return int(series(1/(1 - 10*x + x**2)**Rational(5,2), x, 0, n+1).removeO().coeff(x, n)) # _Aitzaz Imtiaz_, Oct 03 2025

%Y Cf. A000332, A387341.

%Y Cf. A387369.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 27 2025