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G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
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%I #6 Jul 20 2022 08:44:39

%S 1,0,3,-3,1,0,9,-18,21,-15,6,-1,0,22,-56,116,-182,196,-140,64,-17,2,0,

%T 51,-144,496,-1329,2436,-3148,2934,-1971,934,-297,57,-5,0,108,-270,

%U 1680,-7005,18846,-36302,52462,-57914,49060,-31724,15412,-5455,1330,-200,14,0,221,-381,5647,-32760,116068,-298976,591690,-920249,1138052,-1125135,889253,-558740,275744,-104672,29524,-5833,721,-42

%N G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

%C Row sums equal A000108, the Catalan numbers:

%C Sum_{k=0..3*n} T(n,k) = A000108(n) for n >= 0.

%C T(n,3*n) = (-1)^(n-1) * A000108(n-1) for n >= 1 (Catalan numbers).

%C Conjecture: T(n,1) = A000716(n) for n >= 1 (number of partitions of n into parts of 3 kinds).

%C The generating functions of some related sequences are given as follows.

%C (1) A(x,x) = Sum_{n>=0} A355351(n)*x^n.

%C (2) A(x,2*x) = Sum_{n>=0} A355352(n)*x^n.

%C (3) A(x,3*x) = Sum_{n>=0} A355353(n)*x^n.

%C (4) A(x,4*x) = Sum_{n>=0} A355354(n)*x^n.

%C (5) A(x,5*x) = Sum_{n>=0} A355355(n)*x^n.

%C (6) A(x,x^2) = Sum_{n>=0} A355356(n)*x^n.

%C (7) A(x^2,x) = Sum_{n>=0} A355357(n)*x^n.

%C (8) A(x,x*y) = Sum_{n>=0} x^n * Sum_{k=0..n} A355350(n,k) * y^k.

%C (9) 1/A(4*x,-1) = 2*Sum_{n>=0} A268300(n)*x^n.

%C (10) A(x,2) = -Sum_{n>=0} A355871(n)*x^n.

%C SPECIFIC VALUES.

%C (V.1) A(x,y) = -exp(-Pi) at x = exp(-2*Pi) and y = exp(Pi) * Pi^(1/4)/gamma(3/4).

%C (V.2) A(x,y) = -exp(-2*Pi) at x = exp(-4*Pi) and y = exp(2*Pi) * Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.

%C (V.3) A(x,y) = -exp(-3*Pi) at x = exp(-6*Pi) and y = exp(3*Pi) * Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.

%C (V.4) A(x,y) = -exp(-4*Pi) at x = exp(-8*Pi) and y = exp(4*Pi) * Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.

%C (V.5) A(x,y) = -exp(-sqrt(3)*Pi) at x = exp(-2*sqrt(3)*Pi) and y = exp(sqrt(3)*Pi) * gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

%H Paul D. Hanna, <a href="/A355870/b355870.txt">Table of n, a(n) for n = 0..3875</a>

%F G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies:

%F (1) y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

%F (2) y = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^n), by the Jacobi triple product identity.

%e G.f.: A(x,y) = 1/(1-y) + x*(y^3 - 3*y^2 + 3*y)/(1-y)^3 + x^2*(-y^6 + 6*y^5 - 15*y^4 + 21*y^3 - 18*y^2 + 9*y)/(1-y)^5 + x^3*(2*y^9 - 17*y^8 + 64*y^7 - 140*y^6 + 196*y^5 - 182*y^4 + 116*y^3 - 56*y^2 + 22*y)/(1-y)^7 + x^4*(-5*y^12 + 57*y^11 - 297*y^10 + 934*y^9 - 1971*y^8 + 2934*y^7 - 3148*y^6 + 2436*y^5 - 1329*y^4 + 496*y^3 - 144*y^2 + 51*y)/(1-y)^9 + x^5*(14*y^15 - 200*y^14 + 1330*y^13 - 5455*y^12 + 15412*y^11 - 31724*y^10 + 49060*y^9 - 57914*y^8 + 52462*y^7 - 36302*y^6 + 18846*y^5 - 7005*y^4 + 1680*y^3 - 270*y^2 + 108*y)/(1-y)^11 + ...

%e where

%e y = ... + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...

%e also,

%e y = (1 - x*A(x,y))*(1 - 1/A(x,y))*(1-x) * (1 - x^2*A(x,y))*(1 - x/A(x,y))*(1-x^2) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y))*(1-x^3) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y))*(1-x^4) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y))*(1-x^n) * ...

%e This triangle of coefficients T(n,k) of x^n*y^k/(1-y)^(2*n+1) in A(x,y), for k = 0..3*n in row n, begins

%e n = 0: [1];

%e n = 1: [0, 3, -3, 1];

%e n = 2: [0, 9, -18, 21, -15, 6, -1];

%e n = 3: [0, 22, -56, 116, -182, 196, -140, 64, -17, 2];

%e n = 4: [0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5];

%e n = 5: [0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14];

%e n = 6: [0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42];

%e n = 7: [0, 429, -63, 18281, -134985, 594399, -1941037, 4947447, -10062669, 16571700, -22316250, 24716922, -22564425, 16956135, -10435305, 5210319, -2078910, 647565, -151825, 25215, -2646, 132]; ...

%e The rightmost border equals the signed Catalan numbers (A000108) shifted right one place.

%e Column 1 appears to equal A000716 (ignoring the initial term).

%e Example: at y = x, we have the g.f. of A355351:

%e A(x,x) = 1/(1-x) + x*(3*x - 3*x^2 + x^3)/(1-x)^3 + x^2*(9*x - 18*x^2 + 21*x^3 - 15*x^4 + 6*x^5 - x^6)/(1-x)^5 + x^3*(22*x - 56*x^2 + 116*x^3 - 182*x^4 + 196*x^5 - 140*x^6 + 64*x^7 - 17*x^8 + 2*x^9)/(1-x)^7 + ... = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + ... + A355351(n)*x^n + ...

%e where x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,x)^n.

%o (PARI) {T(n,k) = my(A=[1/(1-y)],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));

%o A[#A] = polcoeff( (y - sum(m=-t,t, (-1)^m * x^(m*(m+1)/2) * Ser(A)^m )), #A-1,x)/(1-y)^2);polcoeff(A[n+1]*(1-y)^(2*n+1),k,y)}

%o for(n=0,12, for(k=0,3*n, print1( T(n,k),", "));print(""))

%Y Cf. A000108 (row sums), A355871 (y=2).

%Y Cf. A355350 (related triangle), A355351 (y=x), A355352 (y=2*x), A355353 (y=3*x), A355354 (y=4*x), A355355 (y=5*x), A355356 (y=x^2), A355357 (x=x^2,y=x).

%Y Cf. A355360 (related triangle), A000716.

%K sign,tabf

%O 0,3

%A _Paul D. Hanna_, Jul 19 2022