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 1-9 of 9 results.

A075525 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*t^n/n! = ((1+t)*(1+t^2)*(1+t^3)...)^u.

Original entry on oeis.org

1, 1, 1, 8, 3, 1, 6, 35, 6, 1, 144, 110, 95, 10, 1, 480, 1594, 585, 205, 15, 1, 5760, 8064, 8974, 1995, 385, 21, 1, 5040, 125292, 70252, 35329, 5320, 658, 28, 1, 524160, 684144, 1178540, 392364, 110649, 12096, 1050, 36, 1, 2177280, 14215536, 10683180, 7260560, 1630125, 295113, 24570, 1590, 45, 1
Offset: 1

Views

Author

Vladeta Jovovic, Oct 11 2002

Keywords

Comments

Also the Bell transform of A265024. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Examples

			Triangle begins:
      1;
      1,      1;
      8,      3,     1;
      6,     35,     6,     1;
    144,    110,    95,    10,    1;
    480,   1594,   585,   205,   15,   1;
   5760,   8064,  8974,  1995,  385,  21,  1;
   5040, 125292, 70252, 35329, 5320, 658, 28, 1;
  ...
exp(Sum_{n>0} u*A000593(n)*t^n/n) = 1 + u*t/1! + (u+u^2)*t^2/2! + (8*u+3*u^2+u^3)*t^3/3! + (6*u+35*u^2+6*u^3+u^4)*t^4/4! + ...  - _Seiichi Manyama_, Nov 08 2020.

Links

Crossrefs

Column k=1..3 give A265024, A338787, A338788.
Cf. A000593, A008298.

Programs

  • Maple
    # Adds (1,0,0,0,...) as row 0.
seq(PolynomialTools[CoefficientList](n!*coeff(series(mul((1+z^k)^u, k=1..20),z,20),z,n),u), n=0..9); # Peter Luschny, Jan 26 2016
  • Mathematica
    T[n_, k_] := n! SeriesCoefficient[(Times @@ (1 + t^Range[n]))^u, {t, 0, n}, {u, 0, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2019 *)
  • PARI
    a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)*n/d));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1))) \\ Seiichi Manyama, Nov 08 2020 after Peter Luschny
  • Sage
    # uses[bell_matrix from A264428]
# Adds (1,0,0,0,..) as row 0.
d = lambda n: sum((-1)^(d+1)*n/d for d in divisors(n))
bell_matrix(lambda n: factorial(n)*d(n+1), 9) # Peter Luschny, Jan 26 2016

Formula

Row sums give n!*A000009(n).
From Seiichi Manyama, Nov 08 2020: (Start)
E.g.f.: exp(Sum_{n>0} u*A000593(n)*t^n/n).
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} A000593(k)*T(n-k; u)/(n-k)!, T(0; u) = 1. (End)
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} A000593(i_j)/i_j. - Seiichi Manyama, Nov 09 2020.

A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).

Original entry on oeis.org

1, 2, 8, 24, 144, 960, 5760, 40320, 524160, 4354560, 43545600, 638668800, 6706022400, 99632332800, 2092278988800, 20922789888000, 376610217984000, 9247873130496000, 128047474114560000, 2919482409811968000, 77852864261652480000
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 16 2019

Keywords

Crossrefs

Cf. A002131, A002448, A010050, A015128, A015680, A038048, A054785, A228274, A265024, A330504.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}]

Formula

E.g.f.: -log(theta_4(x)) / 2.
E.g.f.: (1/2) * Sum_{k>=1} log((1 + x^k) / (1 - x^k)).
E.g.f.: log(Product_{k>=1} ((1 + x^k) / (1 - x^k))^(1/2)).
E.g.f.: Sum_{k>=1} x^(2*k - 1) / ((2*k - 1) * (1 - x^(2*k - 1))).
exp(2 * Sum_{n>=1} a(n) * x^n / n!) = g.f. of A015128.
a(n) = (n - 1)! * Sum_{d|n, n/d odd} d.

A330388 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * log(1 + x)^k / (k * (1 - log(1 + x)^k)).

Original entry on oeis.org

1, 0, 7, -37, 338, -2816, 28418, -340334, 5015080, -84244704, 1536606168, -29753884392, 609895549872, -13243687082016, 305507366834832, -7523621131117296, 198844500026698752, -5649686902983730560, 171839087043420258432, -5545292300345590210944
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2019

Keywords

Links

Crossrefs

Cf. A000009, A000593, A008275, A088311, A089064, A265024, A298905, A330354, A330387.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + Log[1 + x]^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Dec 15 2019 *)

Formula

E.g.f.: -Sum_{k>=1} log(1 - log(1 + x)^(2*k - 1)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A298905.
exp(Sum_{n>=1} a(n) * (exp(x) - 1)^n / n!) = g.f. of A000009.
a(n) = Sum_{k=1..n} Stirling1(n,k) * (k - 1)! * A000593(k).
E.g.f.: Sum_{k>=1} log(1 + log(1 + x)^k). - Vaclav Kotesovec, Dec 15 2019
Conjecture: a(n) ~ n! * (-1)^(n+1) * Pi^2 * exp(n) / (24 * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Dec 16 2019

A300515 Expansion of e.g.f. log(Sum_{k>=0} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

0, 1, 0, 1, -3, 7, -24, 130, -748, 4446, -30694, 245586, -2131621, 19850237, -201363613, 2214638141, -26037523804, 325653856386, -4331545709166, 61069238694738, -908488414975896, 14220161323121232, -233746798117055047, 4025924893291859919, -72487584601341680720
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

Comments

Logarithmic transform of A000009.

Examples

			E.g.f.: A(x) = x/1! + x^3/3! - 3*x^4/4! + 7*x^5/5! - 24*x^6/6! + 130*x^7/7! - 748*x^8/8! + ...

Links

Crossrefs

Cf. A000009, A265024, A300512, A300514.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(
      j*a(j)*binomial(n, j)*t(n-j), j=1..n-1)/n))(b)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 07 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[Log[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = PartitionsQ[n] - Sum[k Binomial[n, k] PartitionsQ[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 24}]

Formula

E.g.f.: log(Sum_{k>=0} A000009(k)*x^k/k!).

A347817 E.g.f.: Product_{k>=1} (1 + x^k)^sin(x).

Original entry on oeis.org

1, 0, 2, 3, 40, 80, 1760, 8211, 139256, 763272, 19466578, 147696835, 3372858476, 33370016316, 872184749046, 10340382875655, 289042962136272, 3884706041971728, 118640349946950738, 1911641854423398435, 59577007012206421356, 1086774235381609797540, 37138839666110194130670
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A088311, A265024, A335629, A346841, A347893, A347894, A347898.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^sin(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))

Formula

E.g.f.: exp( sin(x) * Sum_{k>=1} x^k / (k*(1 - x^(2*k))) ). - Ilya Gutkovskiy, Sep 18 2021
E.g.f.: exp( sin(x) * Sum_{k>=1} A000593(k)*x^k/k ). - Seiichi Manyama, Sep 18 2021

A347893 E.g.f.: Product_{k>=1} (1 + x^k)^cos(x).

Original entry on oeis.org

1, 1, 2, 9, 30, 195, 1545, 12474, 95564, 1199397, 14287845, 167518846, 2341450386, 34489552331, 540927170147, 10114629115798, 175935142966408, 3184271322683385, 68623817313870153, 1442553498798565142, 31856896467060026670, 787164874800260366287, 19097783293834170329239
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A265024, A346941, A347817, A347894, A347899.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^cos(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(cos(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))

Formula

E.g.f.: exp( cos(x) * Sum_{k>=1} x^k / (k*(1 - x^(2*k))) ). - Ilya Gutkovskiy, Sep 18 2021
E.g.f.: exp( cos(x) * Sum_{k>=1} A000593(k)*x^k/k ). - Seiichi Manyama, Sep 18 2021

A347894 E.g.f.: Product_{k>=1} (1 + x^k)^tan(x).

Original entry on oeis.org

1, 0, 2, 3, 52, 110, 2690, 11676, 247952, 1434600, 37576168, 296088760, 7698854216, 78083294640, 2187100997328, 27174552638520, 806871808214016, 11698163585372736, 370098862531800000, 6300404006917434624, 208037772410558058624, 4032385785901175122560, 141272996628892396692096
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A265024, A335630, A347774, A347817, A347893, A347900.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^tan(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(tan(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))

Formula

E.g.f.: exp( tan(x) * Sum_{k>=1} x^k / (k*(1 - x^(2*k))) ). - Ilya Gutkovskiy, Sep 18 2021
E.g.f.: exp( tan(x) * Sum_{k>=1} A000593(k)*x^k/k ). - Seiichi Manyama, Sep 18 2021

A330387 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 2, 12, 62, 420, 3782, 40572, 463262, 5708820, 80773622, 1319927532, 23675250062, 447145154820, 8830952572262, 185694817024092, 4246473212654462, 105754322266866420, 2811068529133151702, 78039884046777282252, 2243558766132057764462
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2019

Keywords

Crossrefs

Cf. A000009, A000593, A000629, A008277, A088311, A265024, A305550, A330353, A330388.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) (Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + (Exp[x] - 1)^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Dec 15 2019 *)

Formula

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^(2*k - 1)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A305550.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000009.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * A000593(k).
E.g.f.: Sum_{k>=1} log(1 + (exp(x) - 1)^k). - Vaclav Kotesovec, Dec 15 2019
a(n) ~ n! * Pi^2 / (24 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 15 2019

A265022 Row sums of the Bell transform of the complementary Bell numbers (A264435).

Original entry on oeis.org

1, 1, 0, -2, -1, 12, 20, -113, -430, 1278, 10821, -10234, -317048, -384915, 10352420, 42836466, -340348905, -3180089128, 8045616512, 219303897655, 301713947470, -14401913182942, -84197219028827, 824481606288554, 11426928115546036, -23133559937561187
Offset: 0

Views

Author

Peter Luschny, Dec 01 2015

Keywords

Links

Crossrefs

Cf. A000587 (complementary Bell numbers), A264428, A264435, A265023, A265024.

Programs

  • Mathematica
    Table[Sum[BellY[n, k, BellB[Range[n] - 1, -1]], {k, 0, n}], {n, 0, 30}] (* Vladimir Reshetnikov, Nov 09 2016 *)
  • Sage
    # uses[bell_transform from A264428]
def A265022_list(len):
    uno = [1]*len
    complementary_bell_numbers = [sum((-1)^n*b for (n, b) in enumerate (bell_transform(n, uno))) for n in range(len)]
    return [sum(bell_transform(n, complementary_bell_numbers)) for n in range(len)]
A265022_list(26)
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