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A141753
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Main diagonal of A139755, the table of q-derangement numbers of type A.
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3
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1, 1, 2, 7, 24, 86, 313, 1157, 4325, 16303, 61856, 235917, 903620, 3473381, 13391280, 51761781, 200523644, 778342906, 3026400508, 11785538461, 45959004812, 179444813270, 701422450293, 2744562302533, 10749124666643, 42135320616531
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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) = [q^n] { ([n+1]_q)! * Sum_{m=0..n+1} (-1)^m * q^(m(m-1)/2) / ([m]_q)! }; here, the q-factorial of n is denoted by ([n]_q)! = Product_{j=1..n} (1-q^j)/(1-q).
a(n) ~ c * 4^n / sqrt(Pi*n), where c = A048651^2 = QPochhammer(1/2)^2 = 0.083398563863748521534541808155312... - Vaclav Kotesovec, Aug 30 2023
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PROG
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(PARI) {a(n)=polcoeff(prod(j=1, n+1, (1-q^j)/(1-q))* sum(k=0, n+1, (-1)^k*q^(k*(k-1)/2)/if(k==0, 1, prod(j=1, k, (1-q^j)/(1-q)))), n, q)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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