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Triangle of coefficients in the logarithm of a generalized theta function.
4

%I #32 Mar 12 2017 15:39:25

%S 2,0,-4,0,0,8,0,8,0,-16,0,0,-20,0,32,0,0,0,48,0,-64,0,0,0,0,-112,0,

%T 128,0,0,0,-16,0,256,0,-256,0,0,18,0,72,0,-576,0,512,0,0,0,-40,0,-240,

%U 0,1280,0,-1024,0,0,0,0,88,0,704,0,-2816,0,2048,0,0,0,0,0,-160,0,-1920,0,6144,0,-4096,0,0,0,0,-52,0,208,0,4992,0,-13312,0,8192,0,0,0,0,0,224,0,0,0,-12544,0,28672,0,-16384,0,0,0,0,0,0,-720,0,-1280,0,30720,0,-61440,0,32768,0,0,0,32,0,0,0,1984,0,6144,0,-73728,0,131072,0,-65536

%N Triangle of coefficients in the logarithm of a generalized theta function.

%H Paul D. Hanna, <a href="/A227311/b227311.txt">Table of n, a(n) for n = 1..1081 (rows 0..45 of the flattened triangle).</a>

%F G.f.: A(x,y) = log(1 + 2*Sum_{n>=1} y^n * x^(n^2)).

%F T(n,k) = [x^n*y^k/n] log(1 + 2*Sum_{m>=1} y^m*x^(m^2)), for k=1..n, n>=1.

%F Row sums equal -(-1)^n*(sigma(2*n) - sigma(n)), where sigma(n) is the sum of divisors of n (A000203); see A054785.

%F Column sums are the even numbers: Sum_{n=k..k^2} T(n,k) = 2*k, for k>=1.

%F Sum_{n=k..k^2} T(n,k)*k/n = 1 - (-1)^k, for k>=1.

%F Sum_{k=1..n} T(n,k)*2^k = A227312(n), for n>=1.

%F Element T(n,k) formulas:

%F (1) T(n,k) = 0 if n-k is odd.

%F (2) T(n,n) = -(-2)^n for n>=1.

%F (3) T(n+2,n) = -(n+2)*(-2)^(n-1) for n>=2.

%F (4) T(n^2,n) = 2*n^2 for n>=1, and is the last nonzero element in the n-th column.

%F (5) T((n-2)^2 + 1,n) = -4*((n-2)^2 + 1) for n>=2.

%e G.f.: A(x,y) = 2*y*x - 4*y^2*x^2/2 + 8*y^3*x^3/3 + (8*y^2 - 16*y^4)*x^4/4

%e + (-20*y^3 + 32*y^5)*x^5/5 + (48*y^4 - 64*y^6)*x^6/6

%e + (-112*y^5 + 128*y^7)*x^7/7 + (-16*y^4 + 256*y^6 - 256*y^8)*x^8/8

%e + (18*y^3 + 72*y^5 - 576*y^7 + 512*y^9)*x^9/9 +...

%e where

%e exp(A(x,y)) = 1 + 2*y*x + 2*y^2*x^4 + 2*y^3*x^9 + 2*y^4*x^16 + 2*y^5*x^25 +...

%e Triangle begins:

%e n=1: [2];

%e n=2: [0, -4];

%e n=3: [0, 0, 8];

%e n=4: [0, 8, 0, -16];

%e n=5: [0, 0, -20, 0, 32];

%e n=6: [0, 0, 0, 48, 0, -64];

%e n=7: [0, 0, 0, 0, -112, 0, 128];

%e n=8: [0, 0, 0, -16, 0, 256, 0, -256];

%e n=9: [0, 0, 18, 0, 72, 0, -576, 0, 512];

%e n=10: [0, 0, 0, -40, 0, -240, 0, 1280, 0, -1024];

%e n=11: [0, 0, 0, 0, 88, 0, 704, 0, -2816, 0, 2048];

%e n=12: [0, 0, 0, 0, 0, -160, 0, -1920, 0, 6144, 0, -4096];

%e n=13: [0, 0, 0, 0, -52, 0, 208, 0, 4992, 0, -13312, 0, 8192];

%e n=14: [0, 0, 0, 0, 0, 224, 0, 0, 0, -12544, 0, 28672, 0, -16384];

%e n=15: [0, 0, 0, 0, 0, 0, -720, 0, -1280, 0, 30720, 0, -61440, 0, 32768];

%e n=16: [0, 0, 0, 32, 0, 0, 0, 1984, 0, 6144, 0, -73728, 0, 131072, 0, -65536];

%e n=17: [0, 0, 0, 0, -68, 0, 136, 0, -4896, 0, -21760, 0, 174080, 0, -278528, 0, 131072]; ...

%e Explicitly, the row polynomials begin:

%e n=1: 2*y;

%e n=2: -4*y^2;

%e n=3: 8*y^3;

%e n=4: 8*y^2 - 16*y^4;

%e n=5: -20*y^3 + 32*y^5;

%e n=6: 48*y^4 - 64*y^6;

%e n=7: -112*y^5 + 128*y^7;

%e n=8: -16*y^4 + 256*y^6 - 256*y^8;

%e n=9: 18*y^3 + 72*y^5 - 576*y^7 + 512*y^9;

%e n=10: -40*y^4 - 240*y^6 + 1280*y^8 - 1024*y^10;

%e n=11: 88*y^5 + 704*y^7 - 2816*y^9 + 2048*y^11;

%e n=12: -160*y^6 - 1920*y^8 + 6144*y^10 - 4096*y^12;

%e n=13: -52*y^5 + 208*y^7 + 4992*y^9 - 13312*y^11 + 8192*y^13;

%e n=14: 224*y^6 - 12544*y^10 + 28672*y^12 - 16384*y^14;

%e n=15: -720*y^7 - 1280*y^9 + 30720*y^11 - 61440*y^13 + 32768*y^15;

%e n=16: 32*y^4 + 1984*y^8 + 6144*y^10 - 73728*y^12 + 131072*y^14 - 65536*y^16;

%e n=17: -68*y^5 + 136*y^7 - 4896*y^9 - 21760*y^11 + 174080*y^13 - 278528*y^15 + 131072*y^17; ...

%o (PARI) {T(n,k)=n*polcoeff(polcoeff(log(1 + 2*sum(m=1,sqrtint(n),y^m*x^(m^2))+x*O(x^n)),n,x),k,y)}

%o for(n=1,16,for(k=1,n,print1(T(n,k),", "));print(""))

%Y Cf. A216273, A054785, A227312, A227313.

%K sign,tabl

%O 1,1

%A _Paul D. Hanna_, Jul 06 2013