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A127137 Define an array by Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2, 2k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1). Sequence gives Q(0,n). 8
1, 1, -2, -5, 12, 43, -120, -531, 1680, 8601, -30240, -172965, 665280, 4161555, -17297280, -116658675, 518918400, 3735104625, -17643225600, -134498225925, 670442572800, 5380583766075, -28158588057600, -236759435017875, 1295295050649600, 11364769115001225, -64764752532480000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

FORMULA

See A127080 for e.g.f..

a(n) = (-1)^binomial(n,2)*b(n), where b(2*n) = (2*n)!/n! and b(2*n+1) = 4^n*n!* Sum_{j=0..n} binomial(2*j,j)/8^j. - G. C. Greubel, Jan 30 2020

MAPLE

seq( (-1)^binomial(n, 2)*(`if`(`mod`(n, 2)=0, n!/(n/2)!, 2^(n-1)*((n-1)/2)!*add( binomial(2*j, j)/8^j, j=0..((n-1)/2)) ) ), n=0..30); # G. C. Greubel, Jan 30 2020

MATHEMATICA

Q[0, k_]:= (-1)^Binomial[k, 2]*If[EvenQ[k], k!/(k/2)!, 2^((k-1)/2)*(k)!! Beta[1/2, 1/2, (k+1)/2]/Sqrt[2]]//FullSimplify; Table[Q[0, k], {k, 0, 30}] (* G. C. Greubel, Jan 30 2020 *)

PROG

(PARI) a(n) = (-1)^binomial(n, 2)*if(n%2==0, n!/(n/2)!, 2^(n-1)*((n-1)/2)!*sum( j=0, (n-1)/2, binomial(2*j, j)/8^j));

vector(31, n, a(n-1)) \\ G. C. Greubel, Jan 30 2020

(MAGMA)

function b(n)

  if n mod 2 eq 0 then return Factorial(n)/Gamma(n/2+1);

  else return 2^(n-1)*Gamma((n+1)/2)*(&+[Binomial(2*j, j)/8^j: j in [0..((n-1)/2)]]);

  end if; return b; end function;

[Round((-1)^Binomial(n, 2)*b(n)): n in [0..30]]; // G. C. Greubel, Jan 30 2020

(Sage)

@CachedFunction

def b(k):

    if (mod(k, 2)==0): return factorial(k)/factorial(k/2)

    else: return 2^(k-1)*factorial((k-1)/2)*sum(binomial(2*j, j)/8^j for j in (0..(k-1)/2))

def a(k): return (-1)^binomial(k, 2)*b(k)

[a(n) for n in (0..30)] # G. C. Greubel, Jan 30 2020

CROSSREFS

A001813 interleaved with A090470.

Column 0 of array A127080.

Sequence in context: A009739 A062272 A215789 * A172239 A183758 A334811

Adjacent sequences:  A127134 A127135 A127136 * A127138 A127139 A127140

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Mar 24 2007

EXTENSIONS

Typo in name corrected by G. C. Greubel, Jan 30 2020

STATUS

approved

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Last modified March 2 14:38 EST 2021. Contains 341751 sequences. (Running on oeis4.)