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A051255
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Number of cyclically symmetric transpose complement plane partitions in a 2n X 2n X 2n box.
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11
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1, 1, 2, 11, 170, 7429, 920460, 323801820, 323674802088, 919856004546820, 7434724817843114428, 170943292930264547814443, 11183057455425265737399150652, 2081853548182272792243789109645876
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OFFSET
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0,3
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COMMENTS
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Hankel transform of A006013 without initial term is this sequence without initial term. - Michael Somos, May 15 2022
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REFERENCES
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D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; Eq. (6.15), p. 199 (corrected).
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LINKS
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FORMULA
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a(n) ~ exp(1/72) * GAMMA(1/3)^(2/3) * n^(7/72) * 3^(3*n^2 - 3*n/2 + 11/72) / (A^(1/6) * Pi^(1/3) * 2^(4*n^2 - n - 1/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Feb 28 2015
a(n) = Product_{i=0..n-1} (3i+1) C(6i,2i)/(C(4i+1,2i)*(2i+1)), using [Bressoud, Corrections, p. 199: N8]. - M. F. Hasler, Oct 04 2018
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EXAMPLE
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For n=0 there is the empty partition by convention so a(0)=1. For n=1 there is a single cyclically symmetric transpose complement plane partition in a 2 X 2 X 2 box so a(1)=1.
G.f. = 1 + x + 2*x^2 + 11*x^3 + 170*x^4 + 7429*x^5 + 920460*x^6 + 323801820*x^7 + ... - Michael Somos, May 15 2022
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MAPLE
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A051255 := proc(n) local i; mul((3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!), i=0..n-1); end;
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MATHEMATICA
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a[n_] := Product[(3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!), {i, 0, n-1}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 25 2014 *)
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PROG
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(PARI) a(n)=prod(i=0, n-1, (3*i+1)*(6*i)!*(2*i)!/((4*i)!*(4*i+1)!)); \\ Joerg Arndt, Feb 25 2014
(PARI) A051255(n)=prod(i=0, n-1, (3*i+1)*binomial(6*i, 2*i)/binomial(4*i+1, 2*i)/(2*i+1)) \\ M. F. Hasler, Oct 04 2018
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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