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A227311 Triangle of coefficients in the logarithm of a generalized theta function. 4
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, 128, 0, 0, 0, -16, 0, 256, 0, -256, 0, 0, 18, 0, 72, 0, -576, 0, 512, 0, 0, 0, -40, 0, -240, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1081 (rows 0..45 of the flattened triangle).

FORMULA

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

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.

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

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

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

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

Element T(n,k) formulas:

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

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

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

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

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

EXAMPLE

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

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

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

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

where

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 +...

Triangle begins:

n=1: [2];

n=2: [0, -4];

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Explicitly, the row polynomials begin:

n=1: 2*y;

n=2: -4*y^2;

n=3: 8*y^3;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

PROG

(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)}

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

CROSSREFS

Cf. A216273, A054785, A227312, A227313.

Sequence in context: A200038 A249093 A102392 * A178515 A051517 A289359

Adjacent sequences:  A227308 A227309 A227310 * A227312 A227313 A227314

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Jul 06 2013

STATUS

approved

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Last modified October 18 20:10 EDT 2018. Contains 316325 sequences. (Running on oeis4.)