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A249056
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a(n) = Product_{i=1..floor(n/2)} (2n-i)*(2n+i).
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1
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1, 1, 15, 35, 3780, 9504, 2702700, 7001280, 3809332800, 10027987200, 8896257216000, 23655956688000, 31072794873120000, 83196255098880000, 151687572109032960000, 408177466766125056000, 986280987631626086400000, 2664118997179074699264000
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OFFSET
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0,3
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COMMENTS
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For n>1, a(n) is the product of all the numbers in the interval: [2n-floor(n/2), 2n+floor(n/2)] divided by 2n, their average (see example).
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LINKS
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FORMULA
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a(n) = Product_{i=1..floor(n/2)} (2n-i)*(2n+i).
E.g.f.: 1/4+(3/4)*hypergeom([1/5, 2/5, 3/5, 4/5], [1/3, 1/2, 2/3], (3125/108)*x^2)+x*hypergeom([1/2, 3/5, 4/5, 6/5, 7/5], [2/3, 4/3, 3/2, 3/2], (3125/108)*x^2). - Robert Israel, Oct 19 2014
a(n) ~ 25^n * 15^((n+1)/2) * n^n / (4 * 18^n * exp(n)) if n is even, and a(n) ~ 25^n * 15^(n/2) * n^(n-1) / (2 * 18^n * exp(n)) if n is odd. - Vaclav Kotesovec, Oct 21 2014
Conjecture: D-finite with recurrence +36*n*(3*n-1)*(3*n-2)*(879441252621897*n^2 -3487718196355549*n +3398030540730836)*a(n) -12*(n-1)*(17789803949566350*n^4 -310811840802783855*n^3 +977412572410742017*n^2 -1228202974684130738*n +638585163822021576)*a(n-1) -15*(n-2)*(549650782888685625*n^6 -2729474655610903750*n^5 +4816600358041763875*n^4 -3395195710682405750*n^3 +230889009555801948*n^2 +1176562067066757228*n -715918902620114592)*a(n-2) +5*(5*n-11)*(n-3)*(5*n-14)*(5*n-13)*(5*n-12)*(1976644883285150*n^2 -4064135701440945*n +2004921500186538)*a(n-3)=0. - R. J. Mathar, Jul 27 2022
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EXAMPLE
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a(2) = (3*4*5) / 4 = 3*5 = 15.
a(3) = (5*6*7) / 6 = 5*7 = 35.
a(4) = (6*7*8*9*10) / 8 = 6*7*9*10 = 3780.
a(5) = (8*9*10*11*12) / 10 = 8*9*11*12 = 9504.
a(6) = (9*10*11*12*13*14*15) / 12 = 9*10*11*13*14*15 = 2702700.
a(7) = (11*12*13*14*15*16*17) / 14 = 11*12*13*15*16*17 = 7001280.
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MAPLE
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A249056:=n->mul((2*n-i)*(2*n+i), i=1..floor(n/2)): seq(A249056(n), n=0..20);
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MATHEMATICA
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Table[Product[(2 n - i) (2 n + i), {i, Floor[n/2]}], {n, 0, 20}]
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PROG
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(PARI) vector(100, n, n--; prod(k=1, n\2, (2*n-k)*(2*n+k))) \\ Altug Alkan, Oct 28 2015
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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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