OFFSET
1,3
COMMENTS
Compare g.f. to the identity: Sum_{-oo..+oo} (x - y^n)^(n-1) = 0.
The Lucas triangle, A029635, consists of essentially the same coefficients, but differs in signs and initial term.
FORMULA
G.f. A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1).
G.f. A(x,y) = x * Sum_{-oo..+oo} (x - y^n)^n.
G.f. A(x,y) = x/(1-x) + Sum_{n>=1} (-1)^n*x*y^(n^2)*(2 - x*y^n)/(1 - x*y^n)^(n+1).
G.f. A(x,y) = P(x,y) + Q(x,y) where
P(x,y) = Sum_{n>=0} (x - y^n)^(n+1),
P(x,y) = -1 + Sum_{n>=0} (-1)^n * y^(n*(n-1)) / (1 - x*y^n)^(n+1),
Q(x,y) = Sum_{n>=0} (-1)^n * y^(n*(n+1)) / (1 - x*y^(n+1))^n.
EXAMPLE
G.f. A(x,y) = Sum_{n>=1} x^n * Sum_{k>=0} T(n,k) * y^(k*(n+k-1))
such that A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1).
Explicitly, the g.f. of this array begins:
A(x,y) = x*(1 - 2*y + 2*y^4 - 2*y^9 + 2*y^16 - 2*y^25 + 2*y^36 +...)
+ x^2*(1 - 3*y^2 + 5*y^6 - 7*y^12 + 9*y^20 - 11*y^30 + 13*y^42 +...)
+ x^3*(1 - 4*y^3 + 9*y^8 - 16*y^15 + 25*y^24 - 36*y^35 + 49*y^48 +...)
+ x^4*(1 - 5*y^4 + 14*y^10 - 30*y^18 + 55*y^28 - 91*y^40 + 140*y^54 +...)
+ x^5*(1 - 6*y^5 + 20*y^12 - 50*y^21 + 105*y^32 - 196*y^45 + 336*y^60 +...)
+ x^6*(1 - 7*y^6 + 27*y^14 - 77*y^24 + 182*y^36 - 378*y^50 + 714*y^66 +...)
+ x^7*(1 - 8*y^7 + 35*y^16 - 112*y^27 + 294*y^40 - 672*y^55 + 1386*y^72 +...)
+ x^8*(1 - 9*y^8 + 44*y^18 - 156*y^30 + 450*y^44 - 1122*y^60 + 2508*y^78 +...)
+...
Summing along columns gives the alternate g.f.:
A(x,y) = x/(1-x) + Sum_{n>=1} (-1)^n * x * y^(n^2) * (2 - x*y^n)/(1 - x*y^n)^(n+1).
Note that the coefficient of x in A(x,y) is Jacobi's theta_4 function of y.
Also, the coefficient of x^2 in A(x,y) equals Product_{n>=1} (1 - y^(2*n))^3.
RECTANGULAR ARRAY.
This array of coefficients T(n,k) of x^n * y^(k*(n+k-1)) in A(x,y) begins:
n=1: [1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, ...];
n=2: [1, -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, ...];
n=3: [1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, ...];
n=4: [1, -5, 14, -30, 55, -91, 140, -204, 285, -385, 506, ...];
n=5: [1, -6, 20, -50, 105, -196, 336, -540, 825, -1210, 1716, ...];
n=6: [1, -7, 27, -77, 182, -378, 714, -1254, 2079, -3289, 5005, ...];
n=7: [1, -8, 35, -112, 294, -672, 1386, -2640, 4719, -8008, 13013, ...];
n=8: [1, -9, 44, -156, 450, -1122, 2508, -5148, 9867, -17875, 30888, ...];
n=9: [1, -10, 54, -210, 660, -1782, 4290, -9438, 19305, -37180, 68068, ...]; ...
where row n has g.f.: (1 - z) / (1 + z)^n.
The array has the alternate g.f.: (1 - z) / (1 - x + z).
RELATED SERIES.
We may also write A(x,y) = P(x,y) + Q(x,y) where
P(x,y) = -1 + Sum_{n>=0} (-1)^n * y^(n*(n-1)) / (1 - x*y^n)^(n+1),
Q(x,y) = Sum_{n>=0} (-1)^n * y^(n*(n+1)) / (1 - x*y^(n+1))^n.
These series begin as follows:
P(x,y) = (-1 + y^2 - y^6 + y^12 - y^20 + y^30 - y^42 + y^56 - y^72 +...)
+ x*(1 - 2*y + 3*y^4 - 4*y^9 + 5*y^16 - 6*y^25 + 7*y^36 - 8*y^49 +...)
+ x^2*(1 - 3*y^2 + 6*y^6 - 10*y^12 + 15*y^20 - 21*y^30 + 28*y^42 +...)
+ x^3*(1 - 4*y^3 + 10*y^8 - 20*y^15 + 35*y^24 - 56*y^35 + 84*y^48 +...)
+ x^4*(1 - 5*y^4 + 15*y^10 - 35*y^18 + 70*y^28 - 126*y^40 + 210*y^54 +...)
+ x^5*(1 - 6*y^5 + 21*y^12 - 56*y^21 + 126*y^32 - 252*y^45 + 462*y^60 +...)
+ x^6*(1 - 7*y^6 + 28*y^14 - 84*y^24 + 210*y^36 - 462*y^50 + 924*y^66 +...)
+ x^7*(1 - 8*y^7 + 36*y^16 - 120*y^27 + 330*y^40 - 792*y^55 + 1716*y^72 +...)
+...
Q(x,y) = (1 - y^2 + y^6 - y^12 + y^20 - y^30 + y^42 - y^56 + y^72 +...)
+ x*(-y^4 + 2*y^9 - 3*y^16 + 4*y^25 - 5*y^36 + 6*y^49 - 7*y^64 +...)
+ x^2*(-y^6 + 3*y^12 - 6*y^20 + 10*y^30 - 15*y^42 + 21*y^56 +...))
+ x^3*(-y^8 + 4*y^15 - 10*y^24 + 20*y^35 - 35*y^48 + 56*y^63 +...)
+ x^4*(-y^10 + 5*y^18 - 15*y^28 + 35*y^40 - 70*y^54 + 126*y^70 +...)
+ x^5*(-y^12 + 6*y^21 - 21*y^32 + 56*y^45 - 126*y^60 + 252*y^77 +...)
+ x^6*(-y^14 + 7*y^24 - 28*y^36 + 84*y^50 - 210*y^66 + 462*y^84 +...)
+ x^7*(-y^16 + 8*y^27 - 36*y^40 + 120*y^55 - 330*y^72 + 792*y^91 +...)
+...
PROG
(PARI) { T(n, k) = my(z=x+x*O(x^k)); polcoeff( (1-z)/(1+z)^n, k) }
/* Print as a rectangular array: */
for(n=1, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
/* Print as a triangle: */
for(n=0, 14, for(k=0, n, print1(T(n-k+1, k), ", ")); print(""))
/* Print as a flattened array: */
for(n=0, 14, for(k=0, n, print1(T(n-k+1, k), ", ")); )
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Oct 16 2017
STATUS
approved