A370040 Triangle of coefficients T(n,k) in g.f. A(x,y) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2), for n >= 1, as read by rows.
1, 0, 1, 3, 0, 1, -1, 9, 0, 1, 9, -6, 18, 0, 1, -3, 54, -19, 30, 0, 1, 22, -54, 185, -44, 45, 0, 1, -9, 264, -294, 475, -85, 63, 0, 1, 52, -324, 1463, -1026, 1020, -146, 84, 0, 1, -22, 1127, -2715, 5531, -2781, 1939, -231, 108, 0, 1, 111, -1534, 9648, -13430, 16470, -6384, 3374, -344, 135, 0, 1, -51, 4338, -19005, 51853, -49032, 41567, -13020, 5490, -489, 165, 0, 1
Offset: 1
Examples
G.f.: A(x,y) = x*(1) + x^2*(0 + y) + x^3*(3 + y^2) + x^4*(-1 + 9*y + y^3) + x^5*(9 - 6*y + 18*y^2 + y^4) + x^6*(-3 + 54*y - 19*y^2 + 30*y^3 + y^5) + x^7*(22 - 54*y + 185*y^2 - 44*y^3 + 45*y^4 + y^6) + x^8*(-9 + 264*y - 294*y^2 + 475*y^3 - 85*y^4 + 63*y^5 + y^7) + x^9*(52 - 324*y + 1463*y^2 - 1026*y^3 + 1020*y^4 - 146*y^5 + 84*y^6 + y^8) + x^10*(-22 + 1127*y - 2715*y^2 + 5531*y^3 - 2781*y^4 + 1939*y^5 - 231*y^6 + 108*y^7 + y^9) + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
TRIANGLE.
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins
1;
0, 1;
3, 0, 1;
-1, 9, 0, 1;
9, -6, 18, 0, 1;
-3, 54, -19, 30, 0, 1;
22, -54, 185, -44, 45, 0, 1;
-9, 264, -294, 475, -85, 63, 0, 1;
52, -324, 1463, -1026, 1020, -146, 84, 0, 1;
-22, 1127, -2715, 5531, -2781, 1939, -231, 108, 0, 1;
111, -1534, 9648, -13430, 16470, -6384, 3374, -344, 135, 0, 1;
-51, 4338, -19005, 51853, -49032, 41567, -13020, 5490, -489, 165, 0, 1;
230, -6274, 55413, -128974, 208178, -146098, 92869, -24300, 8475, -670, 198, 0, 1; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..2485
Crossrefs
Programs
-
PARI
/* Generate A(x,y) by use of definition in name */ {T(n,k) = my(A=[0,1]); for(i=1,n, A = concat(A,0); A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + y*Ser(A))^m ) - 1 - (y+2)*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/y ); H=A; polcoeff(A[n+1],k,y)} for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print("")) -
PARI
/* Generate A(x,y) recursively using integration wrt y */ {T(n,k) = my(A = x +x*O(x^n), M=sqrtint(n+1), Q = sum(m=1,M, (-1)^m * x^(m^2)) +x*O(x^n)); for(i=0,n, A = (1/y) * intformal( Q / sum(m=-M,n, (-1)^m * m * (x^m + y*A)^(m-1)), y) +x*O(x^n)); polcoeff(polcoeff(A,n,x),k,y)} for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))
Formula
G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k) * x^n*y^k satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + y*A(x,y))^(n-1) = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + y*A(x,y))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^n)^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^n)^(n+1) = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + y*A(x,y)*x^n)^(n+1) = 0.
(7) A(x,y) = (1/y) * Integral Q(x) / Sum_{n=-oo..+oo} (-1)^n * n * (x^n + y*A(x,y))^(n-1) dy, where Q(x) = Sum_{n>=1} (-1)^n * x^(n^2).
(8) A(x,y=0) = (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3, which is the g.f. of column 0 (A370150) defined at y = 0.
Comments