A115903 -id:A115903 - cp's OEIS frontend

A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

1, -18, 54, 108, 486, 2916, 20412, 157464, 1299078, 11258676, 101328084, 939587688, 8926083036, 86514343272, 852784240824, 8527842408240, 86344404383430, 883760374277460, 9132190534200420, 95167038198509640, 999253901084351220, 10563541240034570040

Offset: 0

Michael Somos, Nov 25 2013

sign

From Ralf Steiner, Apr 04 2017: (Start)
By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:
Sum_{k>=0} a(k)^2/16^k = 2F1(-3/2,-3/2,1,9).
Sum_{k>=0} a(k) / 6^k = -i. (End)

			G.f. = 1 - 18*x + 54*x^2 + 108*x^3 + 486*x^4 + 2916*x^5 + 20412*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..930
R. Steiner, Sums of OEIS-A232546, ResearchGate, 2017. - _Ralf Steiner_, Apr 04 2017

Cf. A000168, A002421, A115903.

a[ n_] := SeriesCoefficient[ (1 - 12 x)^(3/2), {x, 0, n}];
Table[9/Sqrt[Pi] 12^n Gamma[-1/2 + n]/Gamma[2 + n], {n, -1, 20}] (* Ralf Steiner, Apr 01 2017 *)
Flatten[{1, -18, Table[4*3^(n+1)*(2*n-4)!/((n-2)!*n!), {n, 2, 25}]}] (* Vaclav Kotesovec, Apr 02 2017 *)

{a(n) = if( n<0, 0, polcoeff( (1 - 12 * x + x * O(x^n))^(3/2), n))};

0 = a(n+2)*(a(n+1) - 42*a(n)) + 18*a(n+1)*(a(n+1) + 8*a(n)) for all n in Z.
a(n+2) = 54 * A000168(n). a(n) = 3^n * A002421(n). Convolution inverse of A115903.
a(n) = 6*(2*n-5)*a(n-1)/n. - R. J. Mathar, Nov 23 2014
G.f.: 1F0(-3/2;;12x). - R. J. Mathar, Aug 09 2015
For n>=2, a(n) = 4*3^(n+1)*(2*n-4)! / ((n-2)!*n!). - Vaclav Kotesovec, Apr 02 2017
Sum_{k>=0} a(k) / 12^k = 0. - Ralf Steiner, Apr 04 2017

A323277 G.f. = (x/6)( 1/(1-12x)^(3/2) - 1/(1-12*x) ).

Original entry on oeis.org

0, 0, 1, 21, 342, 5049, 70794, 961794, 12792492, 167583249, 2170496898, 27864238950, 355198394484, 4501897295274, 56786420175588, 713416451137956, 8931958558413912, 111495926008783809, 1388178160043508018, 17244120146466623166, 213776181450214477092, 2645421031806169214574

Offset: 0

N. J. A. Sloane, Jan 11 2019

nonn

Evgeniy Krasko, Alexander Omelchenko, Enumeration of r-regular maps on the torus. Part I: Rooted maps on the torus, the projective plane and the Klein bottle. Sensed maps on the torus, Discrete Mathematics (2019) Vol. 342, Issue 2, 584-599. Also arXiv:1709.03225 [math.CO]. See last line of Section 2.

a[n_] := (2^(2n-3) 3^(n-2) ((2n-1)!!/2^(n-1) - (n-1)!))/(n-1)!; a[0] = 0;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 14 2019 *)
CoefficientList[Series[x/6 (1/(1-12x)^(3/2)-1/(1-12x)),{x,0,30}],x] (* Harvey P. Dale, Jun 18 2021 *)

(-n+1)*a(n) +6*(4*n-5)*a(n-1) +72*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jan 17 2019
a(n) = A115903(n-1)/6 - 2*A001021(n-2), n>1. - R. J. Mathar, Jan 17 2019
a(n) = (2^(2*n-3)*3^(n-2)*((2*n-1)!!/2^(n-1) - (n-1)!))/(n-1)!, n>0. - Jean-François Alcover, Feb 14 2019

cp's OEIS Frontend

A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

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A323277 G.f. = (x/6)( 1/(1-12x)^(3/2) - 1/(1-12*x) ).

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cp's OEIS Frontend

A232546 Expansion of (1 - 12*x)^(3/2) in powers of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A323277 G.f. = (x/6)*( 1/(1-12*x)^(3/2) - 1/(1-12*x) ).

Views

Author

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Programs

Mathematica

Formula

A323277 G.f. = (x/6)( 1/(1-12x)^(3/2) - 1/(1-12*x) ).