A370338
Expansion of Product_{n>=1} (1 - 3^(n-1)*x^n) * (1 + 3^(n-1)*x^n)^2.
Original entry on oeis.org
1, 1, 2, 11, 24, 114, 297, 1224, 3240, 13230, 37017, 138510, 407754, 1469664, 4413366, 15717969, 47239200, 163408266, 511758000, 1719152586, 5348422224, 18083342907, 56672868240, 187301066040, 594207370746, 1947548449296, 6185182455792, 20263641256656, 64084643627283
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 24*x^4 + 114*x^5 + 297*x^6 + 1224*x^7 + 3240*x^8 + 13230*x^9 + 37017*x^10 + 138510*x^11 + 407754*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 3*x^2)*(1 + 3*x^2)^2 * (1 - 9*x^3)*(1 + 9*x^3)^2 * (1 - 27*x^4)*(1 + 27*x^4)^2 * ... * (1 - 3^(n-1)*x^n)*(1 + 3^(n-1)*x^n)^2 * ...
Compare A(x) to the series that results from a similar infinite product:
(1 - 3*x)*(1 + 3*x)^2 * (1 - 9*x^2)*(1 + 9*x^2)^2 * (1 - 27*x^3)*(1 + 27*x^3)^2 * (1 - 81*x^4)*(1 + 81*x^4)^2 * ... = 1 + 3*x + 27*x^3 + 729*x^6 + 59049*x^10 + 14348907*x^15 + 10460353203*x^21 + 22876792454961*x^28 + ...
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{a(n) = polcoeff( prod(k=1,n, (1 - 3^(k-1)*x^k) * (1 + 3^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,30, print1(a(n),", "))
A370434
Expansion of Product_{n>=1} (1 - 4^(n-1)*x^n) * (1 + 4^(n-1)*x^n)^2.
Original entry on oeis.org
1, 1, 3, 19, 60, 348, 1216, 6480, 23040, 121152, 445696, 2214912, 8475648, 40796160, 158564352, 754302976, 2949120000, 13694926848, 55180001280, 250151436288, 1008079994880, 4570684063744, 18552497111040, 82564035379200, 339344829186048, 1494986847682560, 6161930523770880
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 60*x^4 + 348*x^5 + 1216*x^6 + 6480*x^7 + 23040*x^8 + 121152*x^9 + 445696*x^10 + 2214912*x^11 + 8475648*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 4*x^2)*(1 + 4*x^2)^2 * (1 - 16*x^3)*(1 + 16*x^3)^2 * (1 - 64*x^4)*(1 + 64*x^4)^2 * ... * (1 - 4^(n-1)*x^n)*(1 + 4^(n-1)*x^n)^2 * ...
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{a(n) = polcoeff( prod(k=1,n, (1 - 4^(k-1)*x^k) * (1 + 4^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,40, print1(a(n),", "))
A370435
Expansion of Product_{n>=1} (1 - 5^(n-1)*x^n) * (1 + 5^(n-1)*x^n)^2.
Original entry on oeis.org
1, 1, 4, 29, 120, 820, 3625, 23400, 105000, 669500, 3075625, 18837500, 89237500, 532500000, 2554062500, 15086640625, 72843750000, 421773437500, 2084812500000, 11834804687500, 58638281250000, 332210205078125, 1656773437500000, 9240966796875000, 46624682617187500, 257479980468750000
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 120*x^4 + 820*x^5 + 3625*x^6 + 23400*x^7 + 105000*x^8 + 669500*x^9 + 3075625*x^10 + 18837500*x^11 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 5*x^2)*(1 + 5*x^2)^2 * (1 - 25*x^3)*(1 + 25*x^3)^2 * (1 - 125*x^4)*(1 + 125*x^4)^2 * ... * (1 - 5^(n-1)*x^n)*(1 + 5^(n-1)*x^n)^2 * ...
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{a(n) = polcoeff( prod(k=1,n, (1 - 5^(k-1)*x^k) * (1 + 5^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,40, print1(a(n),", "))
Showing 1-3 of 3 results.
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