A305199
Expansion of e.g.f. Product_{k>=1} (1 + x^k/k)/(1 - x^k/k).
Original entry on oeis.org
1, 2, 6, 28, 152, 1008, 7756, 67688, 659424, 7123776, 84154224, 1079913888, 14962632384, 222447507072, 3531920599008, 59664827178048, 1067975819206656, 20192760528611328, 402169396496004864, 8414121277765679616, 184498963978904644608, 4231186653661629843456
Offset: 0
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a:=series(mul((1+x^k/k)/(1-x^k/k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
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nmax = 21; CoefficientList[Series[Product[(1 + x^k/k)/(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(1 + (-1)^(k + 1)) x^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A345870
Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k!).
Original entry on oeis.org
1, 2, 6, 26, 126, 742, 5166, 40462, 351742, 3458470, 37425406, 440788702, 5633316574, 77379974518, 1140707915262, 18053421105742, 302414295475134, 5364631473148614, 100769601500958078, 1988246969908681278, 41179474537324087454, 896909297854081874454
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, ((1+x^k)/(1-x^k))^(1/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sum(k=0, N\2, (exp(x^(2*k+1))-1)/(2*k+1)))))
A295833
Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^k/k).
Original entry on oeis.org
1, -1, 3, -11, 47, -279, 2089, -16057, 137409, -1417553, 15656651, -187422531, 2501688463, -34832785831, 529520417217, -8723102543009, 146573712239489, -2670058109819937, 52017332039568019, -1041334898093864443, 22335551258991482991, -502509800119879530551, 11641825391540821682393
Offset: 0
E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...)/((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...) = 1 - x + 3*x^2/2! - 11*x^3/3! + 47*x^4/4! - 279*x^5/5! + 2089*x^6/6! - 16057*x^7/7! + ...
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a:=series(mul((1+x^k)^((-1)^k/k),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
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nmax = 22; CoefficientList[Series[Product[(1 + x^k)^((-1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A295834
Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^(k+1)/k).
Original entry on oeis.org
1, 1, -1, -1, 11, 19, -311, -1919, 20201, 154169, -1363249, -14236289, 140759299, 1213688059, -33239720359, -257577468511, 11707385639249, 119005356808561, -3416942071608929, -43117983466829441, 893917358612502011, 13133282766425234531, -411010168576899605911, -7970128344774479644991
Offset: 0
E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...)/((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...) = 1 + x - x^2/2! - x^3/3! + 11*x^4/4! + 19*x^5/5! - 311*x^6/6! - 1919*x^7/7! + ...
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a:=series(mul((1+x^k)^((-1)^(k+1)/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
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nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((-1)^(k+1)/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A296048
Expansion of e.g.f. Product_{k>=1} ((1 - x^k)/(1 + x^k))^(1/k).
Original entry on oeis.org
1, -2, 2, -4, 32, -128, 496, -2336, 29312, -395776, 3194624, -21951488, 277270528, -4027191296, 38850203648, -739834458112, 19460560584704, -299971773661184, 3169121209090048, -51853341314514944, 1234704403684130816, -30653318499154788352, 658369600764729884672, -10809496145754051313664
Offset: 0
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a:=series(mul(((1-x^k)/(1+x^k))^(1/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
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nmax = 23; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[-2 Sum[Total[Mod[Divisors[k], 2] x^k]/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
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