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Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).
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%I #31 Jul 26 2024 12:35:27

%S 0,1,4,9,19,30,52,70,107,136,191,226,314,352,463,523,664,717,919,964,

%T 1205,1282,1546,1603,1992,2009,2414,2504,2958,2974,3606,3553,4223,

%U 4273,4936,4912,5885,5685,6634,6654,7664,7454,8822,8454,9845

%N Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^3 *Product_{i=1..t} (1-x^i) ).

%H Seiichi Manyama, <a href="/A059820/b059820.txt">Table of n, a(n) for n = 0..1000</a>

%H G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/s/s42andrews.html">Some debts I owe</a>, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

%F a(n) = ( 3*A055507(n-1) + 3*A191831(n) + A191829(n) + 2*sigma_0(n) + 3*sigma(n) + sigma_2(n) )/6. - _Seiichi Manyama_, Jul 26 2024

%p Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=3

%o (PARI) D(x, y, n) = sum(k=1, n-1, sigma(k, x)*sigma(n-k, y));

%o D000(n) = sum(k=1, n-1, sigma(k, 0)*D(0, 0, n-k));

%o a(n) = if(n==0, 0, (3*D(0, 0, n)+3*D(0, 1, n)+D000(n)+2*sigma(n, 0)+3*sigma(n)+sigma(n, 2))/6); \\ _Seiichi Manyama_, Jul 26 2024

%Y Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), A059821(k=4), A059822 (k=5), A059823 (k=6), A059824 (k=7), A059825 (k=8).

%Y Cf. A002133, A065608, A191822, A191832.

%Y Cf. A000203, A001157, A055507, A191829 (Andrews's D_{0,0,0}(n)), A191831 (Andrews's D_{0,1}(n)).

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Feb 24 2001