A129275
Column 1 of triangle A129274; a(n) is the coefficient of q^(n+2) in the squared q-factorial of n+2.
Original entry on oeis.org
1, 10, 71, 474, 3103, 20190, 131204, 853176, 5555674, 36237258, 236763125, 1549496420, 10156512792, 66669881442, 438226458380, 2884072387268, 19002479773355, 125335000366692, 827479642104143, 5468060901435850
Offset: 0
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P:= (1+q)^2: A[0]:= 1:
for n from 1 to 50 do
P:= normal(P * (1-q^(n+2))^2/(1-q)^2);
A[n]:= coeff(P,q,n+2);
od:
seq(A[i],i=0..50); # Robert Israel, Jun 25 2018
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a(n)=polcoeff(prod(i=1,n+2,(1-x^i)/(1-x))^2,n+2)
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))
Showing 1-2 of 2 results.
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