A128545
Triangle, read by rows, where T(n,k) is the coefficient of q^(n*k) in the q-binomial coefficient [2*n, n] for n >= k >= 0.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 8, 5, 1, 1, 7, 18, 18, 7, 1, 1, 11, 39, 58, 39, 11, 1, 1, 15, 75, 155, 155, 75, 15, 1, 1, 22, 141, 383, 526, 383, 141, 22, 1, 1, 30, 251, 867, 1555, 1555, 867, 251, 30, 1, 1, 42, 433, 1860, 4192, 5448, 4192, 1860, 433, 42, 1
Offset: 0
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 5, 8, 5, 1;
1, 7, 18, 18, 7, 1;
1, 11, 39, 58, 39, 11, 1;
1, 15, 75, 155, 155, 75, 15, 1;
1, 22, 141, 383, 526, 383, 141, 22, 1;
1, 30, 251, 867, 1555, 1555, 867, 251, 30, 1;
1, 42, 433, 1860, 4192, 5448, 4192, 1860, 433, 42, 1;
...
A128552
Column 2 of triangle A128545; a(n) is the coefficient of q^(2n+4) in the central q-binomial coefficient [2n+4,n+2].
Original entry on oeis.org
1, 3, 8, 18, 39, 75, 141, 251, 433, 724, 1185, 1892, 2972, 4588, 6981, 10480, 15553, 22821, 33164, 47746, 68163, 96542, 135747, 189550, 262997, 362691, 497339, 678300, 920417, 1242898, 1670688, 2235880, 2979809, 3955422, 5230471, 6891234
Offset: 0
a(2) = 8 because we have: 4+4 = 4+3+1 = 4+2+2 = 4+2+1+1 = 3+3+2 = 3+3+1+1 = 3+2+2+1 = 2+2+2+2. - _Geoffrey Critzer_, Sep 27 2013
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with(combinat): p:= numbpart:
s:= proc(n) s(n):= p(n) +`if`(n>0, s(n-1), 0) end:
a:= n-> p(2*n+4) -2*s(n+1):
seq(a(n), n=0..40); # Alois P. Heinz, Sep 27 2013
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Table[nn=2n;Coefficient[Series[Product[(1-x^(n+i))/(1-x^i),{i,1,n}],{x,0,nn}],x^(2n)],{n,1,37}] (* Geoffrey Critzer, Sep 27 2013 *)
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{a(n)=polcoeff(prod(j=n+3,2*n+4,1-q^j)/prod(j=1,n+2,1-q^j),2*n+4,q)}
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{a(n)=numbpart(2*n+4)-2*sum(k=0,n+1,numbpart(k))} \\ Paul D. Hanna, Feb 06 2013
A128554
Column 4 of triangle A128545; a(n) is the coefficient of q^(4n+16) in the central q-binomial coefficient [2n+8,n+4].
Original entry on oeis.org
1, 7, 39, 155, 526, 1555, 4192, 10465, 24620, 55038, 117966, 243723, 487842, 949446, 1802547, 3346632, 6089910, 10881277, 19121293, 33091141, 56466398, 95105255, 158256685, 260386761, 423932473, 683409993, 1091521679, 1728156294
Offset: 0
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a(n)=polcoeff(prod(j=n+5,2*n+8,1-q^j)/prod(j=1,n+4,1-q^j),4*n+16,q)
Showing 1-3 of 3 results.
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