A292221 Expansion of the exponential generating function (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-3/2))/x.

3, -3, 20, -210, 3024, -55440, 1235520, -32432400, 980179200, -33522128640, 1279935820800, -53970627110400, 2490952020480000, -124903451312640000, 6761440164390912000, -393008709555221760000, 24412776311194951680000, -1613955767240110694400000, 113146793787569865523200000, -8384177419658927035269120000

Offset

0,1

Comments

This gives one half of the z-sequence for the generalized unsigned Lah number Sheffer matrix Lah[4,3] = A292219.

For Sheffer a- and z-sequences see a W. Lang link under A006232 with the references for the Riordan case, and also the present link.

a(n) = (-1)^n * A006963(n+2) for all n >= 0. - Michael Somos, Jul 02 2018

Links

G. C. Greubel, Table of n, a(n) for n = 0..365

Wolfdieter Lang, Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.

Formula

a(n) = [x^n/n!] (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-3/2))/x.

a(0) = 3, a(n) = (-2)^n*Product_{j=1..n} (1 + 2*j)/(n+1) = ((-2)^n/(n+1))*A001147(n+1), n >= 1.

0 = a(n)*(-2880*a(n+2) +2760*a(n+3) +952*a(n+4) +45*a(n+5)) +a(n+1)*(+216*a(n+2) -328*a(n+3) -81*a(n+4) -2*a(n+5)) +a(n+2)*(+12*a(n+3) +2*a(n+4)) for all n>0. - Michael Somos, Jul 02 2018

Example

The sequence z(4,3;n) = 2*a(n) begins: {6, -6, 40, -420, 6048, -110880, 2471040, -64864800, 1960358400, -67044257280, 2559871641600, ...}.

Maple

seq(coeff(series(factorial(n)*(1/2)*(1+4*x)*(1-(1+4*x)^(-3/2))/x, x, n+1), x, n), n=0..20); # Muniru A Asiru, Jul 29 2018

Mathematica

a[ n_] := If[ n < 1, 3 Boole[n == 0], (-2)^n (2 n + 1)!! / (n + 1)]; (* Michael Somos, Jul 02 2018 *)
With[{nn=20}, CoefficientList[Series[1/2 (1+4x) (1-(1+4x)^(-3/2))/x, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Dec 24 2018 *)

Prog

(PARI) {a(n) = if( n<1, 3*(n==0), (-1)^n * (2*n + 1)! / (n + 1)!)}; /* Michael Somos, Jul 02 2018 */
(Magma) [3, -3] cat [(-1)^n*Factorial(2*n+1)/Factorial(n+1): n in [2..30]]; // G. C. Greubel, Jul 28 2018

Crossrefs

Cf. A001147, A006232, A006963, A292219.

Sequence in context: A078431 A346750 A292953 * A112534 A006656 A367995

Adjacent sequences: A292218 A292219 A292220 * A292222 A292223 A292224

Keyword

sign,easy

Author

Wolfdieter Lang, Sep 23 2017

Status

approved