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A352093
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G.f. A(x,y) = lim_{N->infinity} (1 - P(N,x,y))/(2*x)^N, where P(0,x,y) = -y, and P(n+1,x,y) = sqrt(1 - 4*x + 4*x*P(n,x,y)) for n = 0..N-1.
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1
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1, 1, 1, 2, 1, 4, 10, 8, 2, 13, 40, 46, 24, 5, 48, 174, 256, 196, 80, 14, 162, 696, 1286, 1328, 814, 280, 42, 600, 2932, 6400, 8188, 6648, 3404, 1008, 132, 2109, 11824, 30348, 46864, 47582, 32336, 14252, 3696, 429, 7760, 48630, 142352, 256264, 311696, 263844, 154224, 59592, 13728, 1430
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OFFSET
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0,4
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LINKS
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FORMULA
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Diagonal = A000108, the Catalan numbers.
Row sums = 2^(n+1) * A006365(n), which is related to binary tree partitions.
G.f. A(x,y) has the following special values.
(1) A(x=1/8, y) = Pi^2/8 + Sum_{n>=1} y^n * 2^n * gamma(n/2)^2 / (4*n!).
(2) A(x=1/8, y) = Pi^2/8 + (Pi/2)*B(y) + C(y), where
B(y) = Sum_{n>=0} [Product_{k=0..n-1} 2*k+1]^2 * y^(2*n+1) / (2*n+1)!,
C(y) = Sum_{n>=1} [Product_{k=1..n-1} 2*k]^2 * y^(2*n) / (2*n)!.
(3) A(x=1/8, y=1/2) = Pi^2*2/9 = Pi^2/8 + Sum_{n>=1} gamma(n/2)^2 / (4*n!).
(4) A(x=1/8, y=-1/2) = Pi^2/18 = Pi^2/8 + Sum_{n>=1} (-1)^n * gamma(n/2)^2 / (4*n!).
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EXAMPLE
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This triangle of coefficients of x^n*y^k in A(x,y) begins:
1, 1;
1, 2, 1;
4, 10, 8, 2;
13, 40, 46, 24, 5;
48, 174, 256, 196, 80, 14;
162, 696, 1286, 1328, 814, 280, 42;
600, 2932, 6400, 8188, 6648, 3404, 1008, 132;
2109, 11824, 30348, 46864, 47582, 32336, 14252, 3696, 429;
7760, 48630, 142352, 256264, 311696, 263844, 154224, 59592, 13728, 1430;
...
The generating function begins
A(x,y) = (y + 1) + (y^2 + 2*y + 1)*x + (2*y^3 + 8*y^2 + 10*y + 4)*x^2 + (5*y^4 + 24*y^3 + 46*y^2 + 40*y + 13)*x^3 + (14*y^5 + 80*y^4 + 196*y^3 + 256*y^2 + 174*y + 48)*x^4 + (42*y^6 + 280*y^5 + 814*y^4 + 1328*y^3 + 1286*y^2 + 696*y + 162)*x^5 + (132*y^7 + 1008*y^6 + 3404*y^5 + 6648*y^4 + 8188*y^3 + 6400*y^2 + 2932*y + 600)*x^6 + (429*y^8 + 3696*y^7 + 14252*y^6 + 32336*y^5 + 47582*y^4 + 46864*y^3 + 30348*y^2 + 11824*y + 2109)*x^7 + (1430*y^9 + 13728*y^8 + 59592*y^7 + 154224*y^6 + 263844*y^5 + 311696*y^4 + 256264*y^3 + 142352*y^2 + 48630*y + 7760)*x^8 + (4862*y^10 + 51480*y^9 + 248622*y^8 + 723552*y^7 + 1411452*y^6 + 1939152*y^5 + 1912716*y^4 + 1347040*y^3 + 652486*y^2 + 197080*y + 28166)*x^9 + ...
Specific values.
A(x, y=-1) = 0, for all x.
A(x=1/8, y=1/2) = Pi^2*2/9.
A(x=1/8, y=-1/2) = Pi^2/18.
At x = 1/8, the sum along column n is given by
_ Sum_{m>=0} T(m,n)/8^m = 2^n * gamma(n/2)^2 / (4*n!).
Explicitly, at x = 1/8, the sums along columns begin:
Sum_{n>=0} T(n,0)/8^n = Pi^2/8 = 1 + 1/8 + 4/8^2 + 13/8^3 + 48/8^4 + ...;
Sum_{n>=0} T(n,1)/8^n = (Pi/2) = 1 + 2/8 + 10/8^2 + 40/8^3 + 174/8^4 + ...;
Sum_{n>=0} T(n,2)/8^n = 1/2 = 1/8 + 8/8^2 + 46/8^3 + 256/8^4 + 1286/8^5 + ...;
Sum_{n>=0} T(n,3)/8^n = (Pi/2)/3! = 2/8^2 + 24/8^3 + 196/8^4 + 1328/8^5 + ...;
Sum_{n>=0} T(n,4)/8^n = 4/4! = 5/8^3 + 80/8^4 + 814/8^5 + 6648/8^6 + ...;
Sum_{n>=0} T(n,5)/8^n = (Pi/2)*9/5! = 14/8^4 + 280/8^5 + 3404/8^6 + ...;
Sum_{n>=0} T(n,6)/8^n = 64/6! = 42/8^5 + 1008/8^6 + 14252/8^7 + ...;
...
Notice that A(x=1/8, y=-1) = 0 is equivalent to
Pi^2 = Sum_{n>=1} (-2)^(n+1) * gamma(n/2)^2 / n!.
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PROG
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(PARI) /* Prints N Rows of this triangle: */
N = 20;
{T(n, k) = my(P = -y + x*O(x^(2*N+1)));
for(i=1, N+1, P = sqrt(1 - 4*x + 4*x*P +x*O(x^(2*N+1))); );
Axy = (1 - P)/2^(N+1)/x^(N+1); polcoeff(polcoeff(Axy, n, x), k, y)}
for(n=0, N, for(k=0, n+1, print1(T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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