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A109956
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Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.
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3
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1, -1, 1, 3, -4, 1, -12, 18, -7, 1, 55, -88, 42, -10, 1, -273, 455, -245, 75, -13, 1, 1428, -2448, 1428, -510, 117, -16, 1, -7752, 13566, -8379, 3325, -910, 168, -19, 1, 43263, -76912, 49588, -21252, 6578, -1472, 228, -22, 1, -246675, 444015, -296010, 134550, -45630, 11700, -2223, 297, -25, 1
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OFFSET
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0,4
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COMMENTS
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Riordan array (g,f) where f/(1-f)^3=x and g=1-f.
First column is (-1)^n*binomial(3n,n)/(2n+1), a signed version of A001764.
Second column is a signed version of A006629.
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LINKS
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FORMULA
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Number triangle T(n, k) = (-1)^(n-k)*((3k+1)/(2n+k+1))*binomial(3n, n-k).
If u(m,n) = (-1)^n*(Sum_{k=0..n} T(n,k)*((m+1)*k+1)) and v(m,n) = (-1)^n*(Sum_{k=0..n} (-1)^k*T(n,k)*m^k) and D(x) is the g.f. of A001764 then P(m,x) = Sum_{n>=0} u(m,n)*x^n = 1-(m+1)*x*D(x)^2 and Q(m,x) = Sum_{n>=0} v(m,n)*x^n = 1/P(m,x).
If G(k,x) is the g.f. of column k (k>=0) then G(k,x) = G(0,x)^(3*k+1). (End)
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EXAMPLE
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Triangle begins:
1;
-1, 1;
3, -4, 1;
-12, 18, -7, 1;
55, -88, 42, -10, 1;
-273, 455, -245, 75, -13, 1;
...
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MAPLE
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# Function RiordanSquare defined in A321620.
tt := sin(arcsin(3*sqrt(x*3/4))/3)/sqrt(x*3/4): R := RiordanSquare(tt, 11):
seq(seq(LinearAlgebra:-Row(R, n)[k]*(-1)^(n+k), k=1..n), n=1..11); # Peter Luschny, Nov 27 2018
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MATHEMATICA
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T[n_, k_] := (-1)^(n - k)((3k + 1)/(2n + k + 1)) Binomial[3n, n - k];
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PROG
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(PARI) tabl(nn) = {my(m = matrix(nn, nn, n, k, if (n<k, 0, binomial(n+2*k-3, 3*k-3)))); m = 1/m; for (n=1, nn, for (k=1, n, print1(m[n, k], ", "); ); print(); ); } \\ Michel Marcus, Nov 20 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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