A015033
q-Catalan numbers (binomial version) for q=3.
Original entry on oeis.org
1, 1, 10, 847, 627382, 4138659802, 244829520301060, 130191700295480695111, 622829375926755523108996006, 26812578369717035183629988539429726, 10387976772168532331015929118843873280496300
Offset: 0
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q:=3; [1] cat [((1-q)/(1-q^(n+1)))*(&*[(1-q^(2*n-k))/(1-q^(k+1)): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2018
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Table[2 QBinomial[2n, n, 3]/(3^(n+1) - 1), {n, 0, 20}]
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q=3; for(n=0, 20, print1(((1-q)/(1-q^(n+1)))*prod(k=0,n-1, (1-q^(2*n-k))/(1-q^(k+1))), ", ")) \\ G. C. Greubel, Nov 11 2018
A015034
q-Catalan numbers (binomial version) for q=4.
Original entry on oeis.org
1, 1, 17, 4433, 18245201, 1197172898385, 1255709588423576145, 21068918017101222558779985, 5655752483351603939678821837720145, 24291387778773301588924456932322615789898321
Offset: 0
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q:=4; [1] cat [((1-q)/(1-q^(n+1)))*(&*[(1-q^(2*n-k))/(1-q^(k+1)): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2018
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Table[3*QBinomial[2 n, n, 4]/(4^(n + 1) - 1), {n, 0, 20}] (* G. C. Greubel, Nov 11 2018 *)
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q=4; for(n=0, 20, print1(((1-q)/(1-q^(n+1)))*prod(k=0,n-1, (1-q^(2*n-k))/(1-q^(k+1))), ", ")) \\ G. C. Greubel, Nov 11 2018
A136097
a(n) = A135951(n) /[(2^(n+1)-1) * 2^(n*(n-1)/2)].
Original entry on oeis.org
1, -1, 5, -93, 6477, -1733677, 1816333805, -7526310334829, 124031223014725741, -8152285307423733458541, 2140200604371078953284092525, -2245805993494514875022552272042605, 9423041917569791458584837551185555483245
Offset: 0
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Table[(-1)^n QBinomial[2n, n, 2]/(2^(n+1) - 1), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
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a(n)=local(q=2,A=matrix(2*n+1,2*n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1); A[2*n+1,n+1]/( (q^(n+1)-1)/(q-1) * q^(n*(n-1)/2) )
A384437
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th q-Catalan number for q=k.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 5, 1, 1, 1, 10, 93, 14, 1, 1, 1, 17, 847, 6477, 42, 1, 1, 1, 26, 4433, 627382, 1733677, 132, 1, 1, 1, 37, 16401, 18245201, 4138659802, 1816333805, 429, 1, 1, 1, 50, 48205, 256754526, 1197172898385, 244829520301060, 7526310334829, 1430, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 2, 5, 10, 17, 26, ...
1, 5, 93, 847, 4433, 16401, ...
1, 14, 6477, 627382, 18245201, 256754526, ...
1, 42, 1733677, 4138659802, 1197172898385, 100333200992026, ...
Columns k=0..12 give
A000012,
A000108,
A015030,
A015033,
A015034,
A015035,
A015037,
A015038,
A015039,
A015040,
A015041,
A015042,
A015055.
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a(n, k) = if(k==1, binomial(2*n, n)/(n+1), (1-k)/(1-k^(n+1))*prod(j=0, n-1, (1-k^(2*n-j))/(1-k^(j+1))));
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from sage.combinat.q_analogues import q_catalan_number
def a(n, k): return q_catalan_number(n, k)
Showing 1-4 of 4 results.
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