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.

Showing 1-10 of 17 results. Next

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).

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).

A294392 E.g.f.: exp(Sum_{n>=1} A001227(n) * x^n).

Original entry on oeis.org

1, 1, 3, 19, 97, 801, 7411, 73123, 821409, 10977697, 151612291, 2286137811, 38308830913, 669163118209, 12649211055027, 257559356068771, 5432325991339201, 121949878889492673, 2907330680764076419, 71860237654425159187, 1871308081194213959841
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2017

Keywords

Links

Crossrefs

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): this sequence (k=0), A294394 (k=1), A294395 (k=2).
Cf. A001227, A206303.

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSum[k, Mod[#, 2] &]*a[n - k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d%2)*x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001227(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^(2*k-1)/(1 - x^(2*k-1))). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(n*log(n)). - Vaclav Kotesovec, Sep 07 2018

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

A320650 Expansion of 1/(1 - Sum_{k>=1} x^k/(1 - x^(2*k))).

Original entry on oeis.org

1, 1, 2, 5, 10, 22, 48, 103, 222, 481, 1038, 2241, 4842, 10456, 22582, 48776, 105342, 227514, 491386, 1061281, 2292132, 4950510, 10692006, 23092378, 49874474, 107717891, 232646956, 502466304, 1085216744, 2343829586, 5062156694, 10933145610, 23613191032
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 18 2018

Keywords

Comments

Invert transform of A001227.

Links

Crossrefs

Cf. A001227, A129921, A206303.

Programs

  • Maple
    a:=series(1/(1-add(x^k/(1-x^(2*k)),k=1..100)),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 - Sum[x^k/(1 - x^(2 k)), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[Mod[d, 2], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^k)).
G.f.: 1/(1 + x * (d/dx) log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1)))).
a(0) = 1; a(n) = Sum_{k=1..n} A001227(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).

A345871 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/(2*k!)).

Original entry on oeis.org

1, 1, 2, 7, 23, 116, 707, 4775, 34092, 326723, 3255927, 35131570, 404387205, 5178352921, 72399248378, 1124473250815, 17051538263075, 287692287405292, 5225129815132463, 97469631563584567, 1891891950093538380, 40846072052629411027, 895935864764993940483
Offset: 0

Views

Author

Seiichi Manyama, Jun 27 2021

Keywords

Links

Crossrefs

Cf. A005408, A206303, A209902, A298906, A345870.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, ((1+x^k)/(1-x^k))^(1/(2*k!)))))
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\2, (exp(x^(2*k+1))-1)/(2*k+1)))))

Formula

E.g.f.: exp( Sum_{k>=0} (exp(x^(2*k+1)) - 1)/(2*k+1) ).

A362696 Expansion of e.g.f. Product_{k>0} (1 - x^(3*k-2))^(-1/(3*k-2)).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 900, 7020, 62460, 562140, 5984280, 67252680, 863165160, 11700148680, 173098134000, 2625661170000, 45310413258000, 782198417206800, 14310269286746400, 280333959468789600, 6002139207488767200, 129820528515538159200, 2934651197018947982400
Offset: 0

Views

Author

Seiichi Manyama, Jul 07 2023

Keywords

Links

Crossrefs

Cf. A001817, A035382, A206303, A262947, A362697.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1,N, (1-x^(3*k-2))^(1/(3*k-2)))))

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A001817(k) * a(n-k)/(n-k)!.

A362697 Expansion of e.g.f. Product_{k>0} (1 - x^(3*k-1))^(-1/(3*k-1)).

Original entry on oeis.org

1, 0, 1, 0, 9, 24, 225, 504, 16065, 27216, 1555281, 6123600, 159249321, 779262120, 31816914129, 240363179784, 8207359913025, 66059979227424, 2145292484152545, 19782668403572256, 1015331126023222281, 7961977144683689400, 454920488042137314561
Offset: 0

Views

Author

Seiichi Manyama, Jul 07 2023

Keywords

Links

Crossrefs

Cf. A001822, A035386, A206303, A262946, A362696.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^(3*k-1))^(1/(3*k-1)))))

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A001822(k) * a(n-k)/(n-k)!.
Showing 1-10 of 17 results. Next

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