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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 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 January 17 19:58 EST 2019. Contains 319251 sequences. (Running on oeis4.)