A028343 -id:A028343 - cp's OEIS frontend

A294356 E.g.f.: Product_{k>0} (1+x^k)^(-1/k).

1, -1, 1, -5, 23, -119, 619, -4759, 48145, -476657, 4249961, -48286061, 691331431, -9132207655, 117900772963, -2025161870159, 37607411624609, -628236985455329, 10768798391659345, -215626810984559317, 4751529623277906871, -105427459848063440471

Offset: 0

Seiichi Manyama, Oct 29 2017

sign

Cf. A028342, A028343, A168243.

nmax = 20; CoefficientList[Series[Product[1/(1+x^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 29 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1/(1+x^k)^(1/k))))

A295792 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k).

Original entry on oeis.org

1, 2, 6, 28, 152, 1008, 7936, 70208, 689664, 7618816, 92013824, 1202362368, 17053410304, 258928934912, 4197838491648, 72840915607552, 1334630802489344, 25799982480556032, 527187369241870336, 11292834065764450304, 253498950169144590336, 5965951790211865772032, 146341359815078034538496

Offset: 0

Ilya Gutkovskiy, Nov 27 2017

nonn

Convolution of A028342 and A168243. - Vaclav Kotesovec, Sep 07 2018

Vaclav Kotesovec, Table of n, a(n) for n = 0..445

Cf. A001227, A028342, A028343, A054844, A156616, A168243, A206303, A294356.

a:=series(mul(((1+x^k)/(1-x^k))^(1/k),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

nmax = 22; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(2*Sum_{k>=1} A001227(k)*x^k/k).

E.g.f.: exp(Sum_{k>=1} A054844(k)*x^k/k).

A294402 E.g.f.: exp(-Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, -1, -3, -1, 1, 279, 301, 12263, 5601, -431281, -2140739, -77720721, -1755429983, -12569445721, 85768062381, -4458503862121, 43351731658561, 546719071653663, 31735514726673661, 291860504886837599, 5860390638855992001, 208620917963122666679

Offset: 0

Seiichi Manyama, Oct 30 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..448

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294403 (k=1), A294404 (k=2).

Cf. A000005, A018804, A028343, A294363.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j). - Ilya Gutkovskiy, Aug 17 2021

A294463 E.g.f.: Product_{k>0} (1-k*x^k)^(1/k).

Original entry on oeis.org

1, -1, -2, 0, -12, 180, -1080, 15120, -45360, -15120, 6501600, 166320000, -6017457600, 73297224000, 724669545600, -32528399904000, 169180371360000, 6794185638240000, -119705492402496000, 2601008778880512000, -119160456995099520000

Offset: 0

Seiichi Manyama, Oct 31 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

Cf. A028343, A055225, A294462.

N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1-k*x^k)^(1/k))))

a(0) = 1 and a(n) = -(n-1)! * Sum_{k=1..n} A055225(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: exp(-Sum_{k>=1} Sum_{j>=1} j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018

A345751 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k)^(1/k).

Original entry on oeis.org

1, -1, -2, -3, -3, 40, 477, 4375, 45154, 486817, 5002397, 54970652, 732601449, 10046371231, 113632306694, 1051655108629, 12585372336141, 202763995934160, -863641466773595, -247388278229558697, -10810815349601723990, -311011007642247422759

Offset: 0

Seiichi Manyama, Jun 26 2021

sign

Stirling transform of A028343.

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000005, A028343, A048993, A336100, A345750, A345751, A345752.

max = 21; Range[0, max]! * CoefficientList[Series[Product[(1 - (Exp[x] - 1)^k)^(1/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Jun 26 2021 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-(exp(x)-1)^k)^(1/k))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, numdiv(k)*(exp(x)-1)^k/k))))

E.g.f.: exp( -Sum_{k>=1} d(k) * (exp(x) - 1)^k / k ), where d(n) is the number of divisors of n.

a(n) = Sum_{k=0..n} Stirling2(n,k) * A028343(k).

A345762 E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).

Original entry on oeis.org

1, -1, -1, 2, 0, 29, -135, 727, -1967, -6074, 94510, 1548051, -41361089, 408842095, 213929807, -41951737904, 130060640466, 10569226878107, -229371598130229, 3327344803563111, -31418096993670379, -383829978086171112, 17799865170898698140, 220582224147105677385

Offset: 0

Seiichi Manyama, Jun 26 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A028343, A087906, A185895, A209902, A330199, A345758, A346039, A346057.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-x^k)^(1/k!))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, (exp(x^k)-1)/k))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d-1)!)*x^k/k))))

a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, sumdiv(k, d, 1/(d-1)!)*a(n-k)/(n-k)!));

E.g.f.: exp( -Sum_{k>=1} (exp(x^k) - 1)/k ).
E.g.f.: exp( -Sum_{k>=1} A087906(k)*x^k/k! ).
a(n) = -(n-1)! * Sum_{k=1..n} (Sum_{d|k} 1/(d-1)!) * a(n-k)/(n-k)! for n > 0.

A013643 Numbers k such that the continued fraction for sqrt(k) has period 3.

Original entry on oeis.org

41, 130, 269, 370, 458, 697, 986, 1313, 1325, 1613, 1714, 2153, 2642, 2834, 3181, 3770, 4409, 4778, 4933, 5098, 5837, 5954, 6626, 7465, 7610, 8354, 9293, 10282, 10865, 11257, 11321, 12410, 13033, 13549, 14698, 14738, 15977, 17266, 17989

Offset: 1

N. J. A. Sloane, Clark Kimberling, and Walter Gilbert

nonn

All numbers of the form (5n+1)^2 + 4n + 1 for n>0 are elements of this sequence. Numbers of the above form have the continued fraction expansion [5n+1,[2,2,10n+2]]. General square roots of integers with period 3 continued fraction expansions have expansions of the form [n,[2m,2m,2n]]. - David Terr, Jun 15 2004

Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors in this reference!).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from T. D. Noe)

Cf. A044292, A044673, A067896, A028343, A044373, A044754.

cfp3Q[n_]:=Module[{s=Sqrt[n]},If[IntegerQ[s],1,Length[ ContinuedFraction[ s][[2]]]==3]]; Select[Range[18000],cfp3Q] (* Harvey P. Dale, May 30 2019 *)

The general form of these numbers is d = d(m, n) = a^2 + 4mn + 1, where m and n are positive integers and a = a(m, n) = (4m^2 + 1)n + m, for which the continued fraction expansion of sqrt(d) is [a;[2m, 2m, 2a]]. - David Terr, Jul 20 2004

A299033 a(n) = n! * [x^n] Product_{k>=1} (1 - x^k)^(n/k).

Original entry on oeis.org

1, -1, 0, 15, -136, 885, -4896, 43085, -787200, 7775271, 326355200, -22138191801, 781498160640, -18924340012435, 239123351330304, 5915023788331125, -568462201562300416, 25327272129182225295, -795994018378027868160, 15538852668590468027711

Offset: 0

Ilya Gutkovskiy, Feb 01 2018

sign

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 - x^k)^(n/k) begins:
n = 0: (1),  0,    0,    0,     0,      0,      0,  ...
n = 1:  1, (-1),  -1,    1,    -1,     41,   -131,  ...
n = 2:  1,  -2,   (0),   8,    -4,     72,   -704,  ...
n = 3:  1,  -3,    3,  (15),  -45,     63,  -1539,  ...
n = 4:  1,  -4,    8,   16, (-136),   224,  -1856,  ...
n = 5:  1,  -5,   15,    5,  -265,   (885), -2075,  ...
n = 6:  1,  -6,   24,  -24,  -396,   2376, (-4896), ...

Cf. A000005, A028343, A281267, A299034.

Table[n! SeriesCoefficient[Product[(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) = n! * [x^n] exp(-n*Sum_{k>=1} d(k)*x^k/k), where d(k) is the number of divisors of k (A000005).

A294616 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: Product_{j>0} (1-j^k*x^j)^(1/j).

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, -1, -2, 1, 1, -1, -4, 0, -1, 1, -1, -8, -6, -12, 41, 1, -1, -16, -30, -72, 180, -131, 1, -1, -32, -114, -360, 840, -1080, 1499, 1, -1, -64, -390, -1656, 4200, -8640, 15120, -4159, 1, -1, -128, -1266, -7272, 22440, -69120, 161280, -45360

Offset: 0

Seiichi Manyama, Nov 05 2017

			Square array begins:
    1,   1,   1,    1,     1,      1, ...
   -1,  -1,  -1,   -1,    -1,     -1, ...
   -1,  -2,  -4,   -8,   -16,    -32, ...
    1,   0,  -6,  -30,  -114,   -390, ...
   -1, -12, -72, -360, -1656,  -7272, ...
   41, 180, 840, 4200, 22440, 126600, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..1 give A028343, A294463.
Rows n=0..3 give A000012, (-1)*A000012, (-1)*A000079, (-1)*A245804.
Cf. A294761.

A(0,k) = 1 and A(n,k) = -(n-1)! * Sum_{j=1..n} (Sum_{d|j} d^(k*j/d)) * A(n-j,k)/(n-j)! for n > 0.

A304496 Expansion of e.g.f. Product_{k>=1} (1 - x^k)^H(k), where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, -1, -3, -2, 3, 261, 745, 12412, 16289, -260081, -5424199, -96985734, -2047127621, -17402659299, -84365982987, -2937186832544, 39650368238977, 1047895936025183, 35975009604881845, 638531451763185398, 14668256344792565331, 248159858571597211093, 6320237684944085611809

Offset: 0

Ilya Gutkovskiy, May 13 2018

sign

N. J. A. Sloane, Transforms

Cf. A001008, A002805, A028343, A303970, A304494.

nmax = 22; CoefficientList[Series[Product[(1 - x^k)^HarmonicNumber[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d HarmonicNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

E.g.f.: Product_{k>=1} (1 - x^k)^(A001008(k)/A002805(k)).

cp's OEIS Frontend

A294356 E.g.f.: Product_{k>0} (1+x^k)^(-1/k).

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A295792 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k).

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A294402 E.g.f.: exp(-Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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A294463 E.g.f.: Product_{k>0} (1-k*x^k)^(1/k).

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A345751 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k)^(1/k).

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A345762 E.g.f.: Product_{k>=1} (1 - x^k)^(1/k!).

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A013643 Numbers k such that the continued fraction for sqrt(k) has period 3.

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A299033 a(n) = n! * [x^n] Product_{k>=1} (1 - x^k)^(n/k).

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