A352093 - OEIS

%I #9 Mar 05 2022 12:01:05

%S 1,1,1,2,1,4,10,8,2,13,40,46,24,5,48,174,256,196,80,14,162,696,1286,

%T 1328,814,280,42,600,2932,6400,8188,6648,3404,1008,132,2109,11824,

%U 30348,46864,47582,32336,14252,3696,429,7760,48630,142352,256264,311696,263844,154224,59592,13728,1430

%N 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.

%F Column 0 = A351509.

%F Column 1 = A351511.

%F Diagonal = A000108, the Catalan numbers.

%F Row sums = 2^(n+1) * A006365(n), which is related to binary tree partitions.

%F G.f. A(x,y) has the following special values.

%F (1) A(x=1/8, y) = Pi^2/8 + Sum_{n>=1} y^n * 2^n * gamma(n/2)^2 / (4*n!).

%F (2) A(x=1/8, y) = Pi^2/8 + (Pi/2)*B(y) + C(y), where

%F B(y) = Sum_{n>=0} [Product_{k=0..n-1} 2*k+1]^2 * y^(2*n+1) / (2*n+1)!,

%F C(y) = Sum_{n>=1} [Product_{k=1..n-1} 2*k]^2 * y^(2*n) / (2*n)!.

%F (3) A(x=1/8, y=1/2) = Pi^2*2/9 = Pi^2/8 + Sum_{n>=1} gamma(n/2)^2 / (4*n!).

%F (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!).

%e This triangle of coefficients of x^n*y^k in A(x,y) begins:

%e      1,     1;

%e      1,     2,      1;

%e      4,    10,      8,      2;

%e     13,    40,     46,     24,      5;

%e     48,   174,    256,    196,     80,     14;

%e    162,   696,   1286,   1328,    814,    280,     42;

%e    600,  2932,   6400,   8188,   6648,   3404,   1008,   132;

%e   2109, 11824,  30348,  46864,  47582,  32336,  14252,  3696,  429;

%e   7760, 48630, 142352, 256264, 311696, 263844, 154224, 59592, 13728, 1430;

%e    ...

%e The generating function begins

%e 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 + ...

%e Specific values.

%e A(x, y=-1) = 0, for all x.

%e A(x=1/8, y=1/2) = Pi^2*2/9.

%e A(x=1/8, y=-1/2) = Pi^2/18.

%e At x = 1/8, the sum along column n is given by

%e _ Sum_{m>=0} T(m,n)/8^m = 2^n * gamma(n/2)^2 / (4*n!).

%e Explicitly, at x = 1/8, the sums along columns begin:

%e Sum_{n>=0} T(n,0)/8^n = Pi^2/8 = 1 + 1/8 + 4/8^2 + 13/8^3 + 48/8^4 + ...;

%e Sum_{n>=0} T(n,1)/8^n = (Pi/2) = 1 + 2/8 + 10/8^2 + 40/8^3 + 174/8^4 + ...;

%e Sum_{n>=0} T(n,2)/8^n = 1/2 = 1/8 + 8/8^2 + 46/8^3 + 256/8^4 + 1286/8^5 + ...;

%e Sum_{n>=0} T(n,3)/8^n = (Pi/2)/3! = 2/8^2 + 24/8^3 + 196/8^4 + 1328/8^5 + ...;

%e Sum_{n>=0} T(n,4)/8^n = 4/4! = 5/8^3 + 80/8^4 + 814/8^5 + 6648/8^6 + ...;

%e Sum_{n>=0} T(n,5)/8^n = (Pi/2)*9/5! = 14/8^4 + 280/8^5 + 3404/8^6 + ...;

%e Sum_{n>=0} T(n,6)/8^n = 64/6! = 42/8^5 + 1008/8^6 + 14252/8^7 + ...;

%e ...

%e Notice that A(x=1/8, y=-1) = 0 is equivalent to

%e Pi^2 = Sum_{n>=1} (-2)^(n+1) * gamma(n/2)^2 / n!.

%o (PARI) /* Prints N Rows of this triangle: */

%o N = 20;

%o {T(n,k) = my(P = -y + x*O(x^(2*N+1)));

%o for(i=1,N+1, P = sqrt(1 - 4*x + 4*x*P +x*O(x^(2*N+1))););

%o Axy = (1 - P)/2^(N+1)/x^(N+1); polcoeff(polcoeff(Axy,n,x),k,y)}

%o for(n=0,N,for(k=0,n+1, print1(T(n,k),", "));print(""))

%Y Cf. A006365, A351509, A351511, A000108.

%K nonn,tabf

%O 0,4

%A _Paul D. Hanna_, Mar 04 2022