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A178529 Self-convolution square-root of A008977, where A008977(n) = (4n)!/(n!)^4. 6
1, 12, 1188, 170544, 28779300, 5318414640, 1041818334480, 212530940233920, 44671347000417060, 9607097095645249200, 2103954263946309574800, 467599488149125265169600, 105196895958882375628016400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In Narumiya and Shiga on bottom of page 157 the g.f. is given as an integral. On page 158 the square of the g.f. is given as a hypergeometric function. - Michael Somos, Aug 12 2014

REFERENCES

N. Narumiya and H. Shiga, "The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope", Proceedings on Moonshine and related topics (MontrĂ©al, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001.  MR1877764 (2002m:14030)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

a(n) = 4^n/(n!)^2 * Product_{k=0..n-1} (8*k+1)*(8*k+3).

a(n) = 2^(8*n) * GAMMA(n+1/8) * GAMMA(n+3/8) /(GAMMA(1/8)*GAMMA(3/8) *GAMMA(n+1)^2). - Vaclav Kotesovec, Mar 07 2014

a(n) ~ GAMMA(5/8)*GAMMA(7/8) * 2^(8*n-3/2) / (Pi^2 * n^(3/2)). - Vaclav Kotesovec, Mar 07 2014

G.f.: F( 1/8, 3/8, 1; x) = 1 / B(3/8, 5/8) * integral_0^1 (u^5 * (1-u)^3 * (1-x*u))^(-1/8) du. - Michael Somos, Aug 12 2014

Convolution square is A008977. - Michael Somos, Aug 12 2014

EXAMPLE

G.f.: A(x) = 1 + 12*x + 1188*x^2 + 170544*x^3 + 28779300*x^4 +...

A(x)^2 = 1 + 24*x + 2520*x^2 + 369600*x^3 +...+ (4n)!/(n!)^4*x^n +...

MATHEMATICA

Table[4^n/(n!)^2*Product[(8*k + 1)*(8*k + 3), {k, 0, n - 1}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 07 2014 *)

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/8, 3/8, 1, 256 x], {x, 0, n}]; (* Michael Somos, Aug 12 2014 *)

a[ n_] := 256^n / n!^2 Pochhammer[ 1/8, n] Pochhammer[ 3/8, n]; (* Michael Somos, Aug 12 2014 *)

PROG

(PARI) {a(n)=4^n*prod(k=0, n-1, (8*k+1)*(8*k+3))/(n!)^2}

(PARI) {a(n)=polcoeff(sqrt(sum(k=0, n, (4*k)!/(k!)^4*x^k)+x*O(x^n)), n)}

CROSSREFS

Cf. A008977.

Sequence in context: A112580 A229691 A180586 * A201642 A177090 A103269

Adjacent sequences:  A178526 A178527 A178528 * A178530 A178531 A178532

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 23 2010

STATUS

approved

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Last modified May 16 10:20 EDT 2021. Contains 343940 sequences. (Running on oeis4.)