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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A294761 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/(1-j^k*x^j)^(1/j).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 11, 1, 1, 6, 18, 59, 1, 1, 10, 36, 132, 339, 1, 1, 18, 84, 384, 900, 2629, 1, 1, 34, 216, 1296, 3240, 10080, 20677, 1, 1, 66, 588, 4704, 13800, 56880, 93240, 202089, 1, 1, 130, 1656, 17712, 64440, 386640, 635040, 1285200, 2066201
Offset: 0

Views

Author

Seiichi Manyama, Nov 08 2017

Keywords

Examples

			Square array begins:
    1,   1,   1,    1,    1, ...
    1,   1,   1,    1,    1, ...
    3,   4,   6,   10,   18, ...
   11,  18,  36,   84,  216, ...
   59, 132, 384, 1296, 4704, ...

Links

Crossrefs

Columns k=0..1 give A028342, A294462.
Rows n=0-1 give A000012.
Cf. A294616.

Formula

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.

A295794 Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).

Original entry on oeis.org

1, 1, 1, 13, 25, 241, 2761, 14701, 153553, 1903105, 27877681, 263555821, 4788201001, 65083782193, 1040877257785, 24098794612621, 373918687272481, 7393663746307201, 164894196647876833, 3504497611085823565, 81863829346282866361, 2257321249626793901041, 49755091945025205954601
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2017

Keywords

Links

Crossrefs

Cf. A028342, A048272, A168243, A206303, A294363, A294392, A295739.

Programs

  • Maple
    a:=series(mul(exp(x^k/(1+x^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[Exp[x^k/(1 + x^k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x D[Log[Product[(1 + x^k)^(1/k), {k, 1, nmax}]], x]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[-k Sum[(-1)^d, {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp(Sum_{k>=1} A048272(k)*x^k).
E.g.f.: exp(x*f'(x)), where f(x) = log(Product_{k>=1} (1 + x^k)^(1/k)).
a(n) ~ exp(2*sqrt(n*log(2)) - 1/4 - n) * n^(n - 1/4) * log(2)^(1/4) / sqrt(2). - Vaclav Kotesovec, Sep 07 2018

A346546 E.g.f.: Product_{k>=1} 1 / (1 - x^k)^(exp(-x)/k).

Original entry on oeis.org

1, 1, 1, 2, 15, 44, 485, 1854, 25781, 170288, 2477485, 12571140, 435748665, 2049818198, 64651106637, 628176476186, 18837010964105, 93248340364152, 6695745240354169, 33794005826851192, 2549048418922818525, 20209158430316698922, 1138228671555859916609
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 16 2021

Keywords

Comments

Exponential transform of A002744.
The first negative term is a(37) = -2641429247236224246927617458359165366254750.

Crossrefs

Cf. A000005, A002744, A028342, A346545, A346547, A346548.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(Exp[-x]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[-x] Sum[DivisorSigma[0, k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A002744[n_] := Sum[(-1)^(n - k) Binomial[n, k] DivisorSigma[0, k] (k - 1)!, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A002744[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp( exp(-x) * Sum_{k>=1} d(k) * x^k / k ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002744(k) * a(n-k).

A318912 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(mu(k)^2/k), where mu = Möbius function (A008683).

Original entry on oeis.org

1, 1, 3, 11, 53, 309, 2359, 18367, 168489, 1690217, 19416491, 233144691, 3187062493, 44901291421, 700058510943, 11509417045799, 200586478516049, 3680237286827217, 72326917665944659, 1467930587827522267, 31855597406715020421, 718484783876745110021, 16993553696264436052103
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 05 2018

Keywords

Crossrefs

Cf. A001221, A008683, A028342, A034444, A073576, A206303, A318913, A318914.

Programs

  • Maple
    seq(n!*coeff(series(mul(1/(1-x^k)^(mobius(k)^2/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(MoebiusMu[k]^2/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[2^PrimeNu[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[2^PrimeNu[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp(Sum_{k>=1} 2^omega(k)*x^k/k), where omega(k) = number of distinct primes dividing k (A001221).

A318914 Expansion of e.g.f. Product_{p prime, k>=1} 1/(1 - x^(p^k))^(1/(p^k)).

Original entry on oeis.org

1, 0, 1, 2, 15, 44, 475, 2274, 33313, 227240, 2920041, 26754650, 469513231, 4613913732, 85842524755, 1174844041994, 24672317426625, 334246510927184, 7985602649948113, 127351500133158450, 3282809137540001551, 60776696924693716700, 1556379682561575238731, 32568139442090869594802
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 05 2018

Keywords

Crossrefs

Cf. A001222, A023894, A028342, A206303, A318912, A318913.

Programs

  • Maple
    seq(n!*coeff(series(exp(add(bigomega(k)*x^k/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^k)^(Boole[PrimePowerQ[k]]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[PrimeOmega[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[PrimeOmega[k] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} bigomega(k)*x^k/k), where bigomega(k) = number of prime divisors of k counted with multiplicity (A001222).

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

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Examples

			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! + ...

Crossrefs

Cf. A028342, A168243, A206303, A284474, A294356, A295792, A295834.

Programs

  • Maple
    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
  • Mathematica
    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

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Examples

			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! + ...

Crossrefs

Cf. A028342, A168243, A206303, A284467, A294356, A295792, A295833.

Programs

  • Maple
    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
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((-1)^(k+1)/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A307679 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^(1/k).

Original entry on oeis.org

1, 1, 5, 35, 323, 3679, 49819, 781465, 13923545, 277563617, 6118251461, 147715469131, 3875706370315, 109781717161375, 3338229675519803, 108443658227589329, 3747688533281296049, 137273241169036231105, 5311844045472206624005, 216505267421266611639667, 9270689769095765333645651
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 21 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 323*x^4/4! + 3679*x^5/5! + 49819*x^6/6! + 781465*x^7/7! + 13923545*x^8/8! + ...
log(A(x)) = x + 4*x^2/2 + 11*x^3/3 + 27*x^4/4 + 62*x^5/5 + 137*x^6/6 + 296*x^7/7 + 630*x^8/8 + 1326*x^9/9 + ... + A160399(k)*x^k/k + ...

Crossrefs

Cf. A000005, A028342, A103446, A160399, A218482, A307680, A320563.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(Sum_{k>=1} d(k)*x^k/(k*(1 - x)^k)), where d(k) is the number of divisors of k (A000005).
a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A028342(k)*n!/k!.

A318966 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k))^(1/(i*j*k)).

Original entry on oeis.org

1, 1, 5, 21, 165, 1077, 11457, 103905, 1345257, 15834825, 237535389, 3372509709, 59235634125, 979573962429, 19224990899865, 366788042231193, 8019002662543953, 171360055378885905, 4132946756763614133, 97947895990285022085, 2576516749059849502581, 67124117357620005459141
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

Links

Crossrefs

Cf. A000005, A007425, A007426, A028342, A174465, A318413, A318695, A318967.

Programs

  • Maple
    a:=series(mul(mul(mul(1/(1-x^(i*j*k))^(1/(i*j*k)),k=1..21),j=1..50),i=1..50),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Product[Product[Product[1/(1 - x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(tau_3(k)/k), where tau_3 = A007425.
E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} Sum_{j|d} tau(j) ) * x^k/k), where tau = number of divisors (A000005).

A319104 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^(k^2))^(1/k^2).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 900, 6300, 56700, 550620, 5506200, 60568200, 782951400, 10351341000, 144918774000, 2173781610000, 38080340298000, 653540914026000, 12158944705908000, 231019949412252000, 4855314209005260000, 102626845031971260000, 2275136280946849320000, 52328134461777534360000
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

Crossrefs

Cf. A001156, A028342, A046951.

Programs

  • Maple
    seq(n!*coeff(series(mul(1/(1-x^(k^2))^(1/k^2),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^k^2)^(1/k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[#]] &]] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} A046951(k)*x^k/k).
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