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A050983
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de Bruijn's S(4,n).
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7
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1, 14, 786, 61340, 5562130, 549676764, 57440496036, 6242164112184, 698300344311570, 79881547652046140, 9301427008157320036, 1098786921802152516024, 131361675994216221116836, 15863471168011822803270200, 1932252897656224864335299400
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OFFSET
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0,2
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COMMENTS
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a(n) is divisible by (n+1). Prime p divides a(p-1). Prime p>2 divides all a(n) from a((p+1)/2) to a(p-1). - Alexander Adamchuk, Jul 05 2006
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REFERENCES
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N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.
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LINKS
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Table of n, a(n) for n=0..14.
Eric Weisstein's World of Mathematics, Binomial Sums
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FORMULA
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a(n) = Sum_{k=-n..+n} (-1)^k*C(2*n,n+k)^4. - Benoit Cloitre, Mar 02 2005
a(n) = (-1)^n * HypergeometricPFQ[ {-2n, -2n, -2n, -2n}, {1, 1, 1}, -1]. - Alexander Adamchuk, Jul 05 2006
E.g.f.: Sum(n>=0,I^n*x^n/n!^4) * Sum(n>=0,(-I)^n*x^n/n!^4) = Sum(n>=0,a(n)*x^(2*n)/n!^4) where I^2=-1. - Paul D. Hanna, Dec 21 2011
a(n) ~ 0.125 k^(8n+3)/(Pi*n)^(3/2) where k = 2 cos(Pi/8) = A179260. This formula is due to de Bruijn 1958. - Charles R Greathouse IV, Dec 28 2011
Recurrence: a(0) = 1, a(1) = 14, 4 * (n + 1) * (2*n + 1)^3 * (48*n^2 + 162*n + 137) * a(n) + (n + 2)^3 * (2*n + 3) * (48*n^2 + 66*n + 23) * a(n+2) = 2 * (4 * (n + 1)^2 * (2*n + 3)^2 * (408*n^2 + 969*n + 431) - (n + 1) * (2*n + 3) * (69*n + 31) + 57*n + 92) * a(n+1). - Vladimir Reshetnikov, Sep 26 2016
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MATHEMATICA
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Sum[ (-1)^(k+n)Binomial[ 2n, k ]^4, {k, 0, 2n} ]
RecurrenceTable[{a[0] == 1, a[1] == 14, 4 (n + 1) (2 n + 1)^3 (48 n^2 + 162 n + 137) a[n] + (n + 2)^3 (2 n + 3) (48 n^2 + 66 n + 23) a[n + 2] == 2 (4 (n + 1)^2 (2 n + 3)^2 (408 n^2 + 969 n + 431) - (n + 1) (2 n + 3) (69 n + 31) + 57 n + 92) a[n + 1]}, a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 26 2016 *)
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PROG
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(PARI) a(n)=sum(k=0, 2*n, (-1)^(k+n)*binomial(2*n, k)^4) \\ Charles R Greathouse IV, Dec 28 2011
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CROSSREFS
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Cf. A000984, A006480, A050984.
Sequence in context: A208254 A210817 A042519 * A183576 A002429 A064345
Adjacent sequences: A050980 A050981 A050982 * A050984 A050985 A050986
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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