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A127728
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Sum of squared coefficients of q in the q-factorials.
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5
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1, 1, 2, 10, 106, 1930, 53612, 2108560, 111482424, 7625997280, 655331699940, 69110082376388, 8775534280695310, 1320693932817784342, 232459627389638257316, 47311901973588298051380
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Two n-permutations are randomly selected from S_n with replacement. a(n)/(n!)^2 is the probability that they will have the same number of inversions. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 15 2010]
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LINKS
| Eric Weisstein's World of Mathematics, q-Factorial from MathWorld.
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EXAMPLE
| Definition of q-factorial of n:
faq(n) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0)=1;
faq(4) = 1*(1 + q)*(1 + q + q^2)*(1 + q + q^2 + q^3) = 1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6;
then a(n) is the sum of squared coefficients of q:
a(4) = 1^2 + 3^2 + 5^2 + 6^2 + 5^2 + 3^2 + 1^2 = 106.
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MATHEMATICA
| Table[Total[ CoefficientList[Expand[Product[Sum[x^i, {i, 0, m}], {m, 1, n - 1}]], x]^2], {n, 0, 15}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 15 2010]
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PROG
| (PARI) {a(n)=local(faq_n=if(n==0, 1, prod(k=1, n, (1-q^k)/(1-q)))); sum(k=0, n*(n-1)/2, polcoeff(faq_n, k, q)^2)}
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CROSSREFS
| Sequence in context: A000595 A087234 A049538 * A185396 A003222 A003167
Adjacent sequences: A127725 A127726 A127727 * A127729 A127730 A127731
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jan 25 2007
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