A144072
Euler transform of powers of 8.
Original entry on oeis.org
1, 8, 100, 1144, 12906, 141848, 1532276, 16290920, 170938483, 1773107760, 18208004664, 185316171472, 1871103319988, 18756665504080, 186798940872312, 1849265718114736, 18207140415436701, 178355043327697976, 1738966407826985884, 16881111732250394440
Offset: 0
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with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->8^j)(n): seq(a(n), n=0..40);
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nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(8^j), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 14 2015 *)
A144073
Euler transform of powers of 9.
Original entry on oeis.org
1, 9, 126, 1623, 20583, 254493, 3091803, 36974025, 436377771, 5091463423, 58811218362, 673298882775, 7647050353038, 86229872235432, 966019964324004, 10757807941399023, 119146632352548516, 1312935665205028374, 14400230629085596621, 157253909597473608945
Offset: 0
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with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->9^j)(n): seq(a(n), n=0..40);
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nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(9^j), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 14 2015 *)
A292837
Euler transform of powers of 10.
Original entry on oeis.org
1, 10, 155, 2220, 31265, 429502, 5796455, 77009640, 1009734835, 13088591470, 167965714273, 2136403822060, 26958029557805, 337733366170870, 4203655872002815, 52010628718162744, 639999271669543500, 7835602953248681200, 95484165081421513000
Offset: 0
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a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
10^d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..30);
A256105
a(n) = [x^n] 2^(2*n) / Product_{k>=1} (1-x^k)^(2^(-k)).
Original entry on oeis.org
1, 2, 10, 36, 166, 556, 2724, 9000, 41542, 153164, 657644, 2325816, 11020508, 38006264, 164662664, 634362320, 2695771462, 9775537676, 43527018396, 156855914904, 687387270260, 2605392165928, 10799896586616, 40214700074800, 178809945153820, 657023566793400
Offset: 0
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Table[2^(2*n) * SeriesCoefficient[Product[1/(1-x^k)^(2^(-k)),{k,1,n}],{x,0,n}], {n,0,30}]
Table[4^n * (CoefficientList[Series[Exp[Sum[x^k/(2*k*(1-x^k/2)),{k,1,n}]],{x,0,n}],x])[[n+1]],{n,0,30}] (* faster *)
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