A355360 G.f. A(x,y) satisfies: x*y*A(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.
1, 0, 1, 0, 3, 2, 0, 9, 12, 5, 0, 22, 54, 46, 14, 0, 51, 196, 282, 175, 42, 0, 108, 630, 1360, 1365, 666, 132, 0, 221, 1836, 5635, 8190, 6321, 2541, 429, 0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430, 0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862, 0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796
Offset: 0
Examples
G.f.: A(x,y) = 1 + x*y + x^2*(3*y + 2*y^2) + x^3*(9*y + 12*y^2 + 5*y^3) + x^4*(22*y + 54*y^2 + 46*y^3 + 14*y^4) + x^5*(51*y + 196*y^2 + 282*y^3 + 175*y^4 + 42*y^5) + x^6*(108*y + 630*y^2 + 1360*y^3 + 1365*y^4 + 666*y^5 + 132*y^6) + x^7*(221*y + 1836*y^2 + 5635*y^3 + 8190*y^4 + 6321*y^5 + 2541*y^6 + 429*y^7) + x^8*(429*y + 4984*y^2 + 20850*y^3 + 41405*y^4 + 45326*y^5 + 28448*y^6 + 9724*y^7 + 1430*y^8) + x^9*(810*y + 12744*y^2 + 70737*y^3 + 184527*y^4 + 270060*y^5 + 237209*y^6 + 125532*y^7 + 37323*y^8 + 4862*y^9) + x^10*(1479*y + 31050*y^2 + 223652*y^3 + 745745*y^4 + 1404102*y^5 + 1625932*y^6 + 1193116*y^7 + 546039*y^8 + 143650*y^9 + 16796*y^10) + ... where x*y*A(x) = ... - x^10/A(x,y)^5 + 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 + ... also, x*y*A(x)*P(x) = (1 - x*A(x,y))*(1 - 1/A(x,y)) * (1 - x^2*A(x,y))*(1 - x/A(x,y)) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y)) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y)) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y)) * ... TRIANGLE. The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins: n=0: [1]; n=1: [0, 1]; n=2: [0, 3, 2]; n=3: [0, 9, 12, 5]; n=4: [0, 22, 54, 46, 14]; n=5: [0, 51, 196, 282, 175, 42]; n=6: [0, 108, 630, 1360, 1365, 666, 132]; n=7: [0, 221, 1836, 5635, 8190, 6321, 2541, 429]; n=8: [0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430]; n=9: [0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862]; n=10: [0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796]; n=11: [0, 2640, 72560, 667005, 2784110, 6565030, 9646462, 9242178, 5826171, 2349490, 554268, 58786]; n=12: [0, 4599, 163632, 1892670, 9729720, 28161819, 51126740, 61555824, 50308245, 27806065, 10023948, 2143428, 208012]; ... in which column 1 appears to equal A000716, the coefficients in P(x)^3, and column 2 appears to equal twice A023005, the coefficients in P(x)^6, where P(x) is the partition function and begins P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + ... + A000041(n)*x^n + ... Also, the power series expansions of P(x)^3 and P(x)^6 begin P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + 429*x^7 + 810*x^8 + 1479*x^9 + 2640*x^10 + ... + A000716(n)*x^n + ... P(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + 6372*x^7 + 15525*x^8 + 36280*x^9 + 81816*x^10 + ... + A023005(n)*x^n + ... The main diagonal equals the Catalan numbers (A000108), where g.f. C(x) = 1 + x*C(x)^2 begins C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..5150
Crossrefs
Programs
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PARI
{T(n,k) = my(A=[1,y],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9)); A[#A] = polcoeff( x*y*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));polcoeff(A[n+1],k,y)} for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))
Formula
G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies:
(1) x*y*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) -x*y*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x,y)^n.
(3) x*y*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
(4) A(x,y) = B(x, y*A(x,y)) and A(x, y/B(x,y)) = B(x,y) where x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * B(x,y)^n, and B(x,y) is the g.f. of table A355350.
Comments