A370019
Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).
Original entry on oeis.org
1, -4, -16, -48, -384, -2816, -24384, -206336, -1815552, -16189440, -146777856, -1346648064, -12487131136, -116810932224, -1101080592384, -10447586845696, -99706199973888, -956400813293568, -9215587975397376, -89158545637244928, -865730439117078528, -8433936444598677504
Offset: 0
G.f.: A(x) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 - 146777856*x^10 + ...
RELATED SERIES.
The cube of the g.f. A(x) equals the g.f. A370018 which starts as
A(x)^3 = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
The reciprocal of the g.f. A(x) equals the g.f. of A370044, which begins
1/A(x) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + ... + A370044(n)*x^n + ...
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{a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 20, print1(a(n), ", "))
A370044
Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(-1/3).
Original entry on oeis.org
1, 4, 32, 240, 2048, 17920, 163904, 1526784, 14473216, 138743808, 1342326528, 13078851584, 128177979392, 1262257356800, 12481163427840, 123845494105088, 1232601926811648, 12300407336042496, 123037059803447296, 1233275751577944064, 12385053557486911488, 124585853452251328512
Offset: 0
G.f.: A(x) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + 1342326528*x^10 + ...
RELATED SERIES.
The cube of 1/A(x) equals the g.f. A370018 which starts as
1/A(x)^3 = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
and 1/A(x) equals the g.f. of A370019, which begins
1/A(x) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 + ... + A370019(n)*x^n + ...
-
{a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n))^(-1/3);
polcoeff(H=A, n)}
for(n=0, 25, print1(a(n), ", "))
A370045
Expansion of 1 / Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).
Original entry on oeis.org
1, 12, 144, 1552, 16512, 172800, 1803200, 18765312, 195167232, 2028914688, 21089678592, 219201730560, 2278287884288, 23679245377536, 246107817345024, 2557891149933568, 26585106479751168, 276308723697205248, 2871777147680423936, 29847423508786839552, 310215112347152351232
Offset: 0
G.f.: A(x) = 1 + 12*x + 144*x^2 + 1552*x^3 + 16512*x^4 + 172800*x^5 + 1803200*x^6 + 18765312*x^7 + 195167232*x^8 + 2028914688*x^9 + 21089678592*x^10 + ...
RELATED SERIES.
The expansion of 1/A(x) is the following series (A370018)
1/A(x) = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
The cube root of A(x) begins
A(x)^(1/3) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + ... + A370044(n)*x^n + ...
Also, the sixth root of A(x) is an integer series starting as
A(x)^(1/6) = 1 + 2*x + 14*x^2 + 92*x^3 + 742*x^4 + 6188*x^5 + 54956*x^6 + 498584*x^7 + 4625478*x^8 + 43493324*x^9 + 413627172*x^10 + ...
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{a(n) = my(A);A = 1 / sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n));polcoeff(H=A, n)}
for(n=0, 25, print1(a(n), ", "))
Showing 1-3 of 3 results.
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