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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
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
Sequence in context: A200038 A249093 A102392 * A178515 A344874 A051517
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Jul 06 2013
STATUS
approved

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)