A049033 Central factorial numbers: unsigned 2nd subdiagonal of A182867.

1, 56, 4368, 489280, 75851776, 15658639360, 4165906530304, 1390437378293760, 569462999991975936, 280969831084430721024, 164441704270786486861824, 112668650067303149573505024

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn

Examples

			(arcsin x)^6 = x^6 + x^8 + 13/15*x^10 + 139/189*x^12 + ...
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Crossrefs

Cf. A001824, A001825, A002455.

Equals 4^n * A001820(n).

Programs

Maple

A049033 := proc(n)
    arcsin(x)^6;
    coeftayl(%,x=0,2*n+6)*(2*n+6)!/6! ;
end proc:
seq(A049033(n),n=0..20) ; # R. J. Mathar, Jan 20 2025

Formula

E.g.f.: (arcsin x)^6; that is, a_k is the coefficient of x^(2*k+6) in (arcsin x)^6 multiplied by (2*k+6)! and divided by 6!. - Joe Keane (jgk(AT)jgk.org)

(-1)^(n-2)*a(n-2) is the coefficient of x^5 in prod(k=0, 2*n, x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002