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A324304
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a(n) = [y^(n-1)] Product_{k=0..n-2} (n + k*y + n*y^2) for n > 1 with a(1) = 1.
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3
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1, 0, 18, 96, 4300, 81360, 3604342, 128389632, 6704335980, 346778956800, 21896347260084, 1459386186255360, 110117675704707190, 8898156449299703040, 786739773441598071750, 74406732202318884372480, 7565016269351818379826372, 818338704493281924572946432, 94154670956813022045927404464, 11458715042302170139584184320000, 1472412964588453156024745207931636
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) ~ c * n! * (27/4)^n / n^2, where c = 1/(6*Pi*sqrt(3*log(3/2))) = 0.04810181967270783985882272373499905248047631331... - Vaclav Kotesovec, Mar 13 2019, updated Mar 17 2024
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EXAMPLE
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E.g.f.: A(x) = x + 18*x^3/3! + 96*x^4/4! + 4300*x^5/5! + 81360*x^6/6! + 3604342*x^7/7! + 128389632*x^8/8! + 6704335980*x^9/9! + 346778956800*x^10/10! + 21896347260084*x^11/11! + 1459386186255360*x^12/12! + ...
RELATED TRIANGLE.
Triangle A324305 of coefficients in Product_{k=0..n-2} (n + k*y + n*y^2), n >= 1, begins
1;
2, 0, 2;
9, 3, 18, 3, 9;
64, 48, 200, 96, 200, 48, 64;
625, 750, 2775, 2280, 4300, 2280, 2775, 750, 625;
7776, 12960, 46440, 53640, 100584, 81360, 100584, 53640, 46440, 12960, 7776;
117649, 252105, 909979, 1337700, 2594501, 2753415, 3604342, 2753415, 2594501, 1337700, 909979, 252105, 117649; ...
in which the central terms, A324305(n, n-1) for n >= 1, form this sequence.
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MATHEMATICA
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Flatten[{1, Table[Coefficient[Expand[Product[(n + k*y + n*y^2), {k, 0, n-2}]], y^(n-1)], {n, 2, 20}]}] (* Vaclav Kotesovec, Mar 13 2019 *)
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PROG
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(PARI) {A324305(n, k) = polcoeff( prod(j=0, n-2, n + j*y + n*y^2), k, y)}
for(n=1, 25, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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