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 A293600 G.f. A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1), as a flattened rectangular array of coefficients T(n,k) of x^n * y^(k*(n+k-1)) in A(x,y) for n>=1. 3

%I

%S 1,1,-2,1,-3,2,1,-4,5,-2,1,-5,9,-7,2,1,-6,14,-16,9,-2,1,-7,20,-30,25,

%T -11,2,1,-8,27,-50,55,-36,13,-2,1,-9,35,-77,105,-91,49,-15,2,1,-10,44,

%U -112,182,-196,140,-64,17,-2,1,-11,54,-156,294,-378,336,-204,81,-19,2,1,-12,65,-210,450,-672,714,-540,285,-100,21,-2,1,-13,77,-275,660,-1122,1386,-1254,825,-385,121,-23,2,1,-14,90,-352,935,-1782,2508,-2640,2079,-1210,506,-144,25,-2,1,-15,104,-442,1287,-2717,4290,-5148,4719,-3289,1716,-650,169,-27,2

%N G.f. A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1), as a flattened rectangular array of coefficients T(n,k) of x^n * y^(k*(n+k-1)) in A(x,y) for n>=1.

%C Compare g.f. to the identity: Sum_{-oo..+oo} (x - y^n)^(n-1) = 0.

%C The Lucas triangle, A029635, consists of essentially the same coefficients, but differs in signs and initial term.

%F G.f. A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1).

%F G.f. A(x,y) = x * Sum_{-oo..+oo} (x - y^n)^n.

%F 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).

%F G.f. A(x,y) = P(x,y) + Q(x,y) where

%F P(x,y) = Sum_{n>=0} (x - y^n)^(n+1),

%F P(x,y) = -1 + Sum_{n>=0} (-1)^n * y^(n*(n-1)) / (1 - x*y^n)^(n+1),

%F Q(x,y) = Sum_{n>=0} (-1)^n * y^(n*(n+1)) / (1 - x*y^(n+1))^n.

%e G.f. A(x,y) = Sum_{n>=1} x^n * Sum_{k>=0} T(n,k) * y^(k*(n+k-1))

%e such that A(x,y) = Sum_{-oo..+oo} (x - y^n)^(n+1).

%e Explicitly, the g.f. of this array begins:

%e A(x,y) = x*(1 - 2*y + 2*y^4 - 2*y^9 + 2*y^16 - 2*y^25 + 2*y^36 +...)

%e + x^2*(1 - 3*y^2 + 5*y^6 - 7*y^12 + 9*y^20 - 11*y^30 + 13*y^42 +...)

%e + x^3*(1 - 4*y^3 + 9*y^8 - 16*y^15 + 25*y^24 - 36*y^35 + 49*y^48 +...)

%e + x^4*(1 - 5*y^4 + 14*y^10 - 30*y^18 + 55*y^28 - 91*y^40 + 140*y^54 +...)

%e + x^5*(1 - 6*y^5 + 20*y^12 - 50*y^21 + 105*y^32 - 196*y^45 + 336*y^60 +...)

%e + x^6*(1 - 7*y^6 + 27*y^14 - 77*y^24 + 182*y^36 - 378*y^50 + 714*y^66 +...)

%e + x^7*(1 - 8*y^7 + 35*y^16 - 112*y^27 + 294*y^40 - 672*y^55 + 1386*y^72 +...)

%e + x^8*(1 - 9*y^8 + 44*y^18 - 156*y^30 + 450*y^44 - 1122*y^60 + 2508*y^78 +...)

%e +...

%e Summing along columns gives the alternate g.f.:

%e 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).

%e Note that the coefficient of x in A(x,y) is Jacobi's theta_4 function of y.

%e Also, the coefficient of x^2 in A(x,y) equals Product_{n>=1} (1 - y^(2*n))^3.

%e RECTANGULAR ARRAY.

%e This array of coefficients T(n,k) of x^n * y^(k*(n+k-1)) in A(x,y) begins:

%e n=1: [1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, ...];

%e n=2: [1, -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, ...];

%e n=3: [1, -4, 9, -16, 25, -36, 49, -64, 81, -100, 121, ...];

%e n=4: [1, -5, 14, -30, 55, -91, 140, -204, 285, -385, 506, ...];

%e n=5: [1, -6, 20, -50, 105, -196, 336, -540, 825, -1210, 1716, ...];

%e n=6: [1, -7, 27, -77, 182, -378, 714, -1254, 2079, -3289, 5005, ...];

%e n=7: [1, -8, 35, -112, 294, -672, 1386, -2640, 4719, -8008, 13013, ...];

%e n=8: [1, -9, 44, -156, 450, -1122, 2508, -5148, 9867, -17875, 30888, ...];

%e n=9: [1, -10, 54, -210, 660, -1782, 4290, -9438, 19305, -37180, 68068, ...]; ...

%e where row n has g.f.: (1 - z) / (1 + z)^n.

%e The array has the alternate g.f.: (1 - z) / (1 - x + z).

%e RELATED SERIES.

%e We may also write A(x,y) = P(x,y) + Q(x,y) where

%e P(x,y) = -1 + Sum_{n>=0} (-1)^n * y^(n*(n-1)) / (1 - x*y^n)^(n+1),

%e Q(x,y) = Sum_{n>=0} (-1)^n * y^(n*(n+1)) / (1 - x*y^(n+1))^n.

%e These series begin as follows:

%e P(x,y) = (-1 + y^2 - y^6 + y^12 - y^20 + y^30 - y^42 + y^56 - y^72 +...)

%e + x*(1 - 2*y + 3*y^4 - 4*y^9 + 5*y^16 - 6*y^25 + 7*y^36 - 8*y^49 +...)

%e + x^2*(1 - 3*y^2 + 6*y^6 - 10*y^12 + 15*y^20 - 21*y^30 + 28*y^42 +...)

%e + x^3*(1 - 4*y^3 + 10*y^8 - 20*y^15 + 35*y^24 - 56*y^35 + 84*y^48 +...)

%e + x^4*(1 - 5*y^4 + 15*y^10 - 35*y^18 + 70*y^28 - 126*y^40 + 210*y^54 +...)

%e + x^5*(1 - 6*y^5 + 21*y^12 - 56*y^21 + 126*y^32 - 252*y^45 + 462*y^60 +...)

%e + x^6*(1 - 7*y^6 + 28*y^14 - 84*y^24 + 210*y^36 - 462*y^50 + 924*y^66 +...)

%e + x^7*(1 - 8*y^7 + 36*y^16 - 120*y^27 + 330*y^40 - 792*y^55 + 1716*y^72 +...)

%e +...

%e Q(x,y) = (1 - y^2 + y^6 - y^12 + y^20 - y^30 + y^42 - y^56 + y^72 +...)

%e + x*(-y^4 + 2*y^9 - 3*y^16 + 4*y^25 - 5*y^36 + 6*y^49 - 7*y^64 +...)

%e + x^2*(-y^6 + 3*y^12 - 6*y^20 + 10*y^30 - 15*y^42 + 21*y^56 +...))

%e + x^3*(-y^8 + 4*y^15 - 10*y^24 + 20*y^35 - 35*y^48 + 56*y^63 +...)

%e + x^4*(-y^10 + 5*y^18 - 15*y^28 + 35*y^40 - 70*y^54 + 126*y^70 +...)

%e + x^5*(-y^12 + 6*y^21 - 21*y^32 + 56*y^45 - 126*y^60 + 252*y^77 +...)

%e + x^6*(-y^14 + 7*y^24 - 28*y^36 + 84*y^50 - 210*y^66 + 462*y^84 +...)

%e + x^7*(-y^16 + 8*y^27 - 36*y^40 + 120*y^55 - 330*y^72 + 792*y^91 +...)

%e +...

%o (PARI) { T(n,k) = my(z=x+x*O(x^k)); polcoeff( (1-z)/(1+z)^n, k) }

%o /* Print as a rectangular array: */

%o for(n=1,10,for(k=0,10,print1(T(n,k),", "));print(""))

%o /* Print as a triangle: */

%o for(n=0,14,for(k=0,n,print1(T(n-k+1,k),", "));print(""))

%o /* Print as a flattened array: */

%o for(n=0,14,for(k=0,n,print1(T(n-k+1,k),", "));)

%Y Cf. A292929, A293385, A029635.

%K sign,tabl

%O 1,3

%A _Paul D. Hanna_, Oct 16 2017

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Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)