cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

1, -1, 1, -5, 23, -119, 619, -4759, 48145, -476657, 4249961, -48286061, 691331431, -9132207655, 117900772963, -2025161870159, 37607411624609, -628236985455329, 10768798391659345, -215626810984559317, 4751529623277906871, -105427459848063440471
Offset: 0

Views

Author

Seiichi Manyama, Oct 29 2017

Keywords

Crossrefs

Cf. A028342, A028343, A168243.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Ilya Gutkovskiy, Nov 27 2017

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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).

A330352 Expansion of e.g.f. -Sum_{k>=1} log(1 - log(1 + x)^k) / k.

Original entry on oeis.org

1, 1, 0, 10, -68, 818, -9782, 130730, -1835752, 27408672, -438578616, 7697802264, -150743052528, 3293454634416, -78787556904864, 2014008113598432, -54001416897306240, 1504891127666322048, -43527807706621236480, 1311515508480252542208
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 11 2019

Keywords

Links

Crossrefs

Cf. A000005, A002744, A008275, A028342, A089064, A318249, A330351, A330353, A330354, A330493.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[-Sum[Log[1 - Log[1 + x]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]

Formula

E.g.f.: Sum_{i>=1} Sum_{j>=1} log(1 + x)^(i*j) / (i*j).
E.g.f.: log(Product_{k>=1} 1 / (1 - log(1 + x)^k)^(1/k)).
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * tau(k), where tau = A000005.

A356336 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k) )^(1/(1-x)).

Original entry on oeis.org

1, 1, 5, 29, 219, 1949, 20587, 245237, 3289577, 48670973, 788572541, 13849348105, 262283664739, 5317530185889, 114939490137235, 2636612228192969, 63955437488072593, 1634890446576454297, 43920715897460109205, 1236660724225711901749, 36412086992371220561771
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2022

Keywords

Crossrefs

Cf. A000005, A028342, A356297, A356335, A356337.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^(1/k)))^(1/(1-x))))
  • PARI
    a356297(n) = n!*sum(k=1, n, sigma(k, 0)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356297(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356297(k) * binomial(n-1,k-1) * a(n-k).

A299034 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k)^(n/k).

Original entry on oeis.org

1, 1, 8, 93, 1544, 32615, 843264, 25739539, 906373376, 36163950849, 1612483625600, 79458277381901, 4288069172500992, 251520785449249927, 15932801526165085184, 1084003570689331039875, 78835487923639854792704, 6103175938145968656408641, 501114006272655771562911744
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 01 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(n/k) begins:
n = 0: (1), 0,   0,    0,     0,      0,       0,  ...
n = 1:  1, (1),  3,   11,    59,    339,    2629,  ...
n = 2:  1,  2,  (8),  40,   260,   1928,   17056,  ...
n = 3:  1,  3,  15,  (93),  711,   6237,   62901,  ...
n = 4:  1,  4,  24,  176, (1544), 15456,  174784,  ...
n = 5:  1,  5,  35,  295,  2915, (32615), 407725,  ...
n = 6:  1,  6,  48,  456,  5004,  61704, (843264), ...

Links

Crossrefs

Cf. A000005, A028342, A255672, A299033.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[1/(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]

Formula

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).
a(n) ~ c * d^n * n^n, where d = 1.7257974131308983723949107467... and c = 0.693704376971941705824592525... - Vaclav Kotesovec, Sep 08 2018

A303970 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^H(k), where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 1, 5, 26, 199, 1599, 17053, 186276, 2460057, 34226729, 537669401, 8925732958, 163894885735, 3151342927823, 65678713377873, 1437541042260704, 33545591623360881, 819213454875992337, 21170268780829522093, 570252657062810041954, 16139888268919495959911, 475126022355752304699455, 14608848314409377281498213
Offset: 0

Views

Author

Ilya Gutkovskiy, May 03 2018

Keywords

Comments

a(n)/n! is the Euler transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].

Links

Crossrefs

Cf. A001008, A002805, A028342.

Programs

  • Maple
    H:= proc(n) option remember; `if`(n=0, 0, 1/n+H(n-1)) end:
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      H(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
    end:
a:= n-> n!*b(n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 03 2018
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(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}]

Formula

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

A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 1, 5, 23, 179, 1279, 13699, 135085, 1764377, 22527521, 344625461, 5283739471, 94562354875, 1685808248383, 33947023942259, 694786150879829, 15613612524749489, 357353282848083265, 8880505496901812197, 224851013929747732231, 6106205671049245677251, 169523515381173773551871
Offset: 0

Views

Author

Ilya Gutkovskiy, May 26 2018

Keywords

Comments

a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Links

Crossrefs

Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.

Programs

  • Maple
    with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018

A318695 Expansion of e.g.f. Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(1/(i*j)).

Original entry on oeis.org

1, 1, 4, 16, 106, 658, 6088, 51952, 592828, 6577948, 88213744, 1173121024, 18663391096, 289030343704, 5157010548064, 92428084599232, 1848308567352592, 37038307949425168, 822602470902709312, 18285742807660340992, 444405771941314880416, 10883864256927386369056, 286778106663948874858624
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 31 2018

Keywords

Links

Crossrefs

Cf. A000005, A006171, A007425, A028342, A280540, A305127, A318696, A318977.

Programs

  • Maple
    seq(n!*coeff(series(mul(1/(1-x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[Product[1/(1 - x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau(k)/k), where tau = number of divisors (A000005).
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} tau(d) ) * x^k/k).

A318913 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k))^(1/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 385, 1854, 18193, 153656, 1924641, 17123930, 276117721, 2880135972, 51150361249, 738748900694, 11608748988705, 198747251005424, 4029150617813953, 67937635488741426, 1607525018948543401, 32739373317847964060, 757174325538283357761, 16444280000832495199982
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 05 2018

Keywords

Crossrefs

Cf. A000607, A001221, A028342, A206303, A318912, A318914.

Programs

  • Maple
    seq(n!*coeff(series(mul(1/(1-x^ithprime(k))^(1/ithprime(k)),k=1..100),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[1/(1 - x^Prime[k])^(1/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[PrimeNu[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[PrimeNu[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-isprime(k)*x^k)^(1/k)))) \\ Seiichi Manyama, Feb 28 2022
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, omega(k)*x^k/k)))) \\ Seiichi Manyama, Feb 28 2022

Formula

E.g.f.: exp(Sum_{k>=1} omega(k)*x^k/k), where omega(k) = number of distinct primes dividing k (A001221).

A151954 Expansion of Product_{k>0} (1-k^2*x^k)^(-1/k).

Original entry on oeis.org

1, 1, 3, 6, 16, 27, 79, 126, 331, 632, 1436, 2509, 6800, 11218, 26044, 51958, 114941, 205183, 502228, 875545, 2027193, 3963938, 8389190, 15504996, 37555290, 66502859, 145809046, 292860564, 621638120, 1156065731, 2701045579
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jul 03 2009

Keywords

Links

Crossrefs

Cf. A028342, A073705, A077335.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[(1-k^2*x^k)^(-1/k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 05 2017 *)

Formula

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} A073705(k)*a(n-k) for n > 0. - Seiichi Manyama, Nov 05 2017
From Vaclav Kotesovec, Nov 05 2017: (Start)
a(n) ~ c * 3^(2*n/3) / n^(2/3), where
c = 4.674336739118905298732313884863019... if mod(n,3)=0
c = 4.299861572054701010776554223312792... if mod(n,3)=1
c = 4.239106098573472870377481583112857... if mod(n,3)=2
(End)
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