A337970 - OEIS

%I #9 Oct 06 2020 19:49:44

%S 6,36,216,776,3876,18576,83376,399376,2034126,9884376,49034376,

%T 245689376,1221921876,6107609376,30563759376,152599609376,

%U 762979296876,3815571921876,19073708984376,95369005859376,476858422109376,2384189755859376,11920935693359376,59605116212109376,298023413134765626

%N L.g.f.: -log( Sum_{n=-oo..+oo} (-p)^n * (p*x)^(n^2) ) = Sum_{n>=1} a(n) * x^n/n, where p = sqrt(5).

%H Paul D. Hanna, <a href="/A337970/b337970.txt">Table of n, a(n) for n = 1..500</a>

%F L.g.f.: -log( Sum_{n>=0} (-1)^n * A337971(n) * x^(n^2) ) = Sum_{n>=1} a(n) * x^n/n, where A337971(n) = 5^(n*(n-1)/2) + 5^(n*(n+1)/2) for n>0 with A337971(0) = 1.

%F L.g.f.: -log( Product_{n>=1} (1 - 5^n*x^(2*n)) * (1 - 5^n*x^(2*n-1)) * (1 - 5^(n-1)*x^(2*n-1)) ) = Sum_{n>=1} a(n) * x^n/n, by the Jacobi triple product identity.

%F L.g.f.: Sum_{n>=1} Sum_{k>=1} ( 5^(k*n)*x^(2*k*n) + (5^n+1)*5^((k-1)*n)*x^((2*k-1)*n) )/n = Sum_{n>=1} a(n) * x^n/n.

%F a(n) = 6 (mod 10).

%e L.g.f.: L(x) = 6*x + 36*x^2/2 + 216*x^3/3 + 776*x^4/4 + 3876*x^5/5 + 18576*x^6/6 + 83376*x^7/7 + 399376*x^8/8 + 2034126*x^9/9 + 9884376*x^10/10 + 49034376*x^11/11 + 245689376*x^12/12 + 1221921876*x^13/13 + 6107609376*x^14/14 + 30563759376*x^15/15 + 152599609376*x^16/16 + ... + a(n)*x^n/n + ...

%e where

%e exp(-L(x)) = 1 - 6*x + 130*x^4 - 15750*x^9 + 9781250*x^16 - 30527343750*x^25 + 476867675781250*x^36 + ... + (-1)^n*A337971(n)*x^(n^2) + ...

%o (PARI) /* By Definition: */

%o {a(n) = n*polcoeff( -log( sum(m=-sqrtint(2*n+1),sqrtint(2*n+1), (-1)^m*5^(m*(m+1)/2)*x^(m^2) +x*O(x^n)) ),n)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) /* By the Jacobi Triple Product identity: */

%o {a(n) = n*polcoeff( -log( prod(m=1,n\2+1, (1 - 5^m*x^(2*m)) * (1 - 5^m*x^(2*m-1)) * (1 - 5^(m-1)*x^(2*m-1)) +x*O(x^n))),n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A337971, A337948, A337950, A337968.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Oct 04 2020