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A208446 Inverse binomial transform of the Franel numbers (A000172). 3
1, 1, 7, 31, 175, 991, 5881, 35617, 219871, 1376095, 8710537, 55644337, 358198369, 2320792657, 15120204295, 98984058271, 650725327231, 4293779332927, 28425752310361, 188739799967425, 1256510215733185, 8385127334900305, 56078904057164215, 375796823748323215 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Starting at offset 1, is this the same as A172634?

LINKS

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

FORMULA

G.f.: Sum_{n>=0} (3*n)!/n!^3 * x^(2*n)*(1+x)^n / (1-x)^(3*n+1).

Equals the binomial transform of A002898.

a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000172(k), where A000172(k) = Sum_{j=0..k} binomial(k,j)^3 forms the Franel numbers.

Recurrence: n^2*a(n) = (2*n-1)^2*a(n-1) + 19*(n-1)^2*a(n-2) + 14*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 7^(n+1)*sqrt(3)/(12*Pi*n). - Vaclav Kotesovec, Oct 20 2012

G.f.: hypergeom([1/3, 1/3],[1],-27*x*(x+1)^2/((1-7*x)^2*(1+2*x)))/((1+2*x)^(1/3)*(1-7*x)^(2/3)). - Mark van Hoeij, May 07 2013

EXAMPLE

G.f.: A(x) = 1 + x + 7*x^2 + 31*x^3 + 175*x^4 + 991*x^5 + 5881*x^6 +...

G.f.: A(x) = 1/(1-x) + 6*x^2*(1+x)/(1-x)^4 + 90*x^4*(1+x)^2/(1-x)^7 + 1680*x^6*(1+x)^3/(1-x)^10 + 34650*x^8*(1+x)^4/(1-x)^13 +...+ A006480(n)*x^(2*n)*(1+x)^n/(1-x)^(3*n+1) +...

MATHEMATICA

Table[SeriesCoefficient[Sum[(3*k)!/k!^3*x^(2*k)*(1+x)^k/(1-x)^(3*k+1), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)

PROG

(PARI) {a(n)=polcoeff(sum(m=0, n, (3*m)!/m!^3*x^(2*m)*(1+x)^m/(1-x+x*O(x^n))^(3*m+1)), n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A000172, A006480.

Sequence in context: A188233 A264608 A172634 * A139151 A139060 A223144

Adjacent sequences:  A208443 A208444 A208445 * A208447 A208448 A208449

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 26 2012

STATUS

approved

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Last modified September 21 15:27 EDT 2017. Contains 292305 sequences.