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A063419
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Central sextinomial coefficients.
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6
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1, 6, 146, 4332, 135954, 4395456, 144840476, 4836766584, 163112472594, 5542414273884, 189456975899496, 6507792553644256, 224442843729333276, 7766945604528200460, 269557528994032024080, 9378595792117360310832
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OFFSET
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0,2
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COMMENTS
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Largest coefficient of (Sum_{j=0..5} x^j)^(2*n). a(n)= A018901(2*n).
The exponent of 2 in prime factorization of a(n) for n=1,2,...,7 is in A023520. - Roman Witula, Oct 07 2012
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REFERENCES
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Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 605, 606.
R. Witula and D. Slota, Central trinomial coefficients and convolution type identities, Congressus Numerantium 201 (2010), 109-126.
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LINKS
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FORMULA
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a(n) = A063260(2*n, 5*n)= [x^(5*n)](Sum_{j=0..5} x^j)^(2*n).
a(n) = Sum_{k=0..floor(5*n/6)} ((-1)^(k)*binomial(2*n,k)*binomial(7*n-6*k-1, 2*n-1)). - Warut Roonguthai, May 22 2006
2*Pi*a(n) = Integral[-Pi;Pi] (sin(6*x)/sin(x))^(2*n) dx. The proof of this fact is in the Witula/Slota paper. - Roman Witula, Oct 07 2012
The Almkvist-Zeilberger algorithm in EKHAD establishes the following recurrence: -31104*(2*n+5)*(2*n+3)*(2*n+1)*(7*n+19)*(5*n+11)*(7*n+20)*(7*n+13)*(n+2)*(n+1)*a(n)+ 864*(7*n+20)*(2*n+5)*(2*n+3)*(n+2)*(25480*n^5+ 223496*n^4+755066*n^3+1223233*n^2+946889*n+279936)*a(n+1)- 6*(5*n+6)*(2*n+5)*(7*n+6)*(499359*n^6+ 6777015*n^5+38079431*n^4+113390385*n^3+18872398*n^2+ 166469280*n+60800544)*a(n+2)+ 5*(5*n+14)*(5*n+13)*(5*n+12)*(7*n+12)*(5*n+11)*(5*n+6)*(7*n+13)*(7*n+6)*(n+3)*a(n+3) = 0. - Doron Zeilberger, Apr 02 2013
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MATHEMATICA
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Table[Sum[(-1)^k*Binomial[2*n, k]*Binomial[7*n - 6*k - 1, 2*n - 1], {k, 0, Floor[5*n/6]}], {n, 0, 50}] (* G. C. Greubel, Mar 02 2017 *)
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PROG
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(PARI) concat([1], for(n=1, 25, print1(sum(k=0, floor(5*n/6), (-1)^(k)*binomial(2*n, k)* binomial(7*n-6*k-1, 2*n-1)), ", "))) \\ G. C. Greubel, Mar 02 2017
(GAP) Concatenation([1], List([1..15], n->Sum([0..Int(5*n/6)], k->(-1)^k*Binomial(2*n, k)*Binomial(7*n-6*k-1, 2*n-1)))); # Muniru A Asiru, Sep 26 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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