A128565
Column 1 of triangle A128564; a(n) equals the number of permutations of {1..n+2} with [n/2+1] inversions for n>=0.
Original entry on oeis.org
1, 2, 5, 9, 29, 49, 174, 285, 1068, 1717, 6655, 10569, 41926, 66013, 266338, 416687, 1703027, 2651355, 10947079, 16976806, 70673825, 109256095, 457927079, 706071989, 2976282415, 4579020513, 19395654894, 29784426945, 126688273871, 194231327451, 829176461458
Offset: 0
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{a(n)=polcoeff(prod(j=1, n+2, (1-q^j)/(1-q)),(n+2)\2,q)}
A098558
Expansion of e.g.f. (1+x)/(1-x).
Original entry on oeis.org
1, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000
Offset: 0
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[1] cat [2*Factorial(n): n in [1..30]]; // G. C. Greubel, Jan 17 2018
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Join[{1}, 2*Range[30]!] (* G. C. Greubel, Jan 17 2018 *)
With[{nn=30},CoefficientList[Series[(1+x)/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 05 2021 *)
a[n_] := Hypergeometric2F1Regularized[1, -n, 2 - n, -1];
Table[a[n], {n, 0, 22}] (* Peter Luschny, Apr 26 2024 *)
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concat([1], vector(30, n, 2*n!)) \\ G. C. Greubel, Jan 17 2018
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CF = ComplexBallField(100)
def a(n):
return Integer(CF(-1).hypergeometric([1, -n], [2 - n], regularized=True))
print([a(n) for n in range(23)]) # Peter Luschny, Apr 26 2024
A128566
Number of permutations of {1..n} with n inversions.
Original entry on oeis.org
1, 0, 0, 1, 5, 22, 90, 359, 1415, 5545, 21670, 84591, 330121, 1288587, 5032235, 19664205, 76893687, 300895513, 1178290263, 4617369760, 18106447251, 71048746505, 278966179936, 1095987764828, 4308300939450, 16944940572831, 66680029591816, 262519664110588
Offset: 0
Diagonal of
A008302 (Mahonian numbers).
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a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..n), q, n+1), q, n):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 05 2013
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Table[SeriesCoefficient[QPochhammer[x, x, n]/(1-x)^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 13 2016 *)
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{a(n)=polcoeff(prod(j=1, n, (1-q^j)/(1-q)),n,q)}
A129276
Triangle, read by rows, where T(n,k) is the coefficient of q^(nk-k) in the squared q-factorial of n.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 8, 8, 1, 1, 42, 106, 42, 1, 1, 241, 1558, 1558, 241, 1, 1, 1444, 23589, 53612, 23589, 1444, 1, 1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1, 1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1
Offset: 0
Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 4 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, (42), 71, 96, (106), 96, 71, (42), 19, 6, (1)],
where the terms enclosed in parenthesis form row 4.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 8, 8, 1;
1, 42, 106, 42, 1;
1, 241, 1558, 1558, 241, 1;
1, 1444, 23589, 53612, 23589, 1444, 1;
1, 8867, 360499, 1747433, 1747433, 360499, 8867, 1;
1, 55320, 5530445, 54794622, 111482424, 54794622, 5530445, 55320, 1; ...
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faq[n_, q_] := Product[(1-q^k)/(1-q), {k, 1, n}]; t[0, 0] = t[1, 0] = t[1, 1] = 1; t[n_, k_] := SeriesCoefficient[faq[n, q]^2, {q, 0, (n-1)*k}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 26 2013 *)
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T(n,k)=if(n==0,1,polcoeff(prod(i=1,n,(1-x^i)/(1-x))^2,(n-1)*k))
A129274
Triangle, read by rows, where T(n,k) is the coefficient of q^(nk+k) in the squared q-factorial of n+1.
Original entry on oeis.org
1, 1, 1, 1, 10, 1, 1, 71, 71, 1, 1, 474, 1930, 474, 1, 1, 3103, 40096, 40096, 3103, 1, 1, 20190, 739929, 2108560, 739929, 20190, 1, 1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1, 1, 853176, 215022825, 3286786158, 7625997280
Offset: 0
Definition of q-factorial of n:
faq(n,q) = Product_{k=1..n} (1-q^k)/(1-q) for n>0, with faq(0,q)=1.
Obtain row 3 from coefficients in the squared q-factorial of 4:
faq(4,q)^2 = 1*(1 + q)^2*(1 + q + q^2)^2*(1 + q + q^2 + q^3)^2
= (1 + 3*q + 5*q^2 + 6*q^3 + 5*q^4 + 3*q^5 + q^6)^2;
the resulting coefficients of q are:
[(1), 6, 19, 42, (71), 96, 106, 96, (71), 42, 19, 6, (1)],
where the terms enclosed in parenthesis form row 3.
Triangle begins:
1;
1, 1;
1, 10, 1;
1, 71, 71, 1;
1, 474, 1930, 474, 1;
1, 3103, 40096, 40096, 3103, 1;
1, 20190, 739929, 2108560, 739929, 20190, 1;
1, 131204, 12836959, 88638236, 88638236, 12836959, 131204, 1;
1, 853176, 215022825, 3286786158, 7625997280, 3286786158, 215022825, 853176, 1; ...
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T(n,k)=polcoeff(prod(i=1,n+1,(1-x^i)/(1-x))^2,(n+1)*k)
Showing 1-5 of 5 results.
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