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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A354309 Expansion of e.g.f. 1/(1 - 2*x)^(x/2).

Original entry on oeis.org

1, 0, 2, 6, 44, 360, 3744, 46200, 662864, 10838016, 198943200, 4050937440, 90613710912, 2208677328000, 58265734055424, 1653914478303360, 50263564166365440, 1628300694034022400, 56012708047907510784, 2039053421375533094400, 78314004507947110456320
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A066166, A354310.
Cf. A053491, A354311, A354315, A354319.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-2x)^(x/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 10 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x)^(x/2)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*2^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 2^(k-2)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-2*k) * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n)). - Vaclav Kotesovec, Mar 14 2024

A356795 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 68, 330, 7674, 73080, 1883440, 28281960, 818625960, 17120406600, 557507325000, 15014517495120, 548643259812816, 18056683281775320, 736892260092195840, 28579282973977498560, 1295028345251832359616, 57666859088090317591680
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A066166, A355842, A356796.
Cf. A356786.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*log(1-x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*log(1-x))/(2*x*log(1-x)))^(1/2)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(2 * x * log(1-x))/2 ).
E.g.f.: A(x) = ( LambertW(2 * x * log(1-x))/(2 * x * log(1-x)) )^(1/2).

A353891 Expansion of e.g.f. exp( (x * log(1-x))^2 / 4 ).

Original entry on oeis.org

1, 0, 0, 0, 6, 30, 165, 1050, 8932, 86184, 909360, 10393020, 129313206, 1743627600, 25314159780, 393346535400, 6512022804960, 114430467296880, 2127154061337480, 41703621476302800, 859966710771029040, 18606040434320713920, 421427283751799685360
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A066166, A353892, A353893.
Cf. A347001, A353880, A353894.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*log(1-x))^2/4)))
  • PARI
    a(n) = n!*sum(k=0, n\4, (2*k)!*abs(stirling(n-2*k, 2*k, 1))/(4^k*k!*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (2*k)! * |Stirling1(n-2*k,2*k)|/(4^k * k! * (n-2*k)!).

A353892 Expansion of e.g.f. exp( -(x * log(1-x))^3 / 36 ).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 20, 210, 1960, 18900, 194880, 2166780, 26172080, 342599400, 4835694864, 73208215080, 1183011385920, 20318534134080, 369549843420384, 7094851788127680, 143377043010268800, 3042204544957939200, 67621161484919380800, 1571319471977711258880
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A066166, A353891, A353893.
Cf. A347002, A353881, A353895.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(x*log(1-x))^3/36)))
  • PARI
    a(n) = n!*sum(k=0, n\6, (3*k)!*abs(stirling(n-3*k, 3*k, 1))/(36^k*k!*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/6)} (3*k)! * |Stirling1(n-3*k,3*k)|/(36^k * k! * (n-3*k)!).

A353893 Expansion of e.g.f. exp( (x * log(1-x))^4 / 576 ).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 70, 1260, 17850, 242550, 3350655, 48108060, 724403680, 11478967500, 191601229820, 3367499575440, 62253354650760, 1208755315895400, 24611454394536780, 524613603866302440, 11687734234226039220, 271715852337632107020
Offset: 0

Views

Author

Seiichi Manyama, May 09 2022

Keywords

Crossrefs

Cf. A066166, A353891, A353892.
Cf. A347003, A353882, A353896.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*log(1-x))^4/576)))
  • PARI
    a(n) = n!*sum(k=0, n\8, (4*k)!*abs(stirling(n-4*k, 4*k, 1))/(576^k*k!*(n-4*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/8)} (4*k)! * |Stirling1(n-4*k,4*k)|/(576^k * k! * (n-4*k)!).

A354310 Expansion of e.g.f. 1/(1 - 3*x)^(x/3).

Original entry on oeis.org

1, 0, 2, 9, 84, 990, 14754, 264600, 5549424, 133217784, 3601384200, 108249692760, 3580724721672, 129250420556400, 5055196156459344, 212951257371183240, 9612027759287831040, 462798880374787387200, 23675607840207619145664, 1282413928716141429168000
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A066166, A354309.
Cf. A351735, A354316, A354320.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x)^(x/3)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, i, j*3^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=2..n} k * 3^(k-2)/(k-1) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-2*k) * |Stirling1(n-k,k)|/(n-k)!.

A354623 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^x.

Original entry on oeis.org

1, 0, 2, 9, 44, 390, 2754, 32760, 310064, 4244184, 54887400, 818615160, 12909921672, 225872515440, 4045885572624, 79360837887240, 1649832369335040, 35666417240193600, 822291935260976064, 19830352438530840960, 501144432316767688320, 13229590606682042436480
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A355064, A356554.
Cf. A000203, A053529, A066166, A356335, A356565.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k)^x))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * sigma(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A356554 Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^k )^x.

Original entry on oeis.org

1, 0, 2, 15, 92, 930, 8514, 116760, 1445744, 23020200, 373858920, 6756785640, 130982295432, 2751191997840, 61046788571664, 1445520760702200, 36387213668348160, 960383111961228480, 26780931923301572544, 781864626481646405760, 23925584882896903854720
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Cf. A354623, A355064.
Cf. A001157, A066166, A356337, A356566.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-x^k)^k)^x))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1, 2)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * sigma_2(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).

A356796 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^3).

Original entry on oeis.org

1, 0, 2, 3, 92, 450, 14454, 141540, 4980128, 78711696, 3048567480, 68677353360, 2930551701384, 86832573553440, 4079649847428960, 150444517302424800, 7768028697749806080, 342721736137376184960, 19392702029822685015360, 994397473912386435004800
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A066166, A355842, A356795.
Cf. A356787.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*log(1-x))/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*log(1-x))/(3*x*log(1-x)))^(1/3)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (3*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(3 * x * log(1-x))/3 ).
E.g.f.: A(x) = ( LambertW(3 * x * log(1-x))/(3 * x * log(1-x)) )^(1/3).

A066165 Variant of Stanley's children's game. Class of n (named) children forms into rings of at least two with exactly one child inside each ring. a(n) gives number of possibilities, including clockwise order (or which hand is held), in each ring.

Original entry on oeis.org

3, 8, 30, 234, 1680, 13040, 119448, 1212120, 13412520, 161968872, 2118607920, 29813747040, 449227822680, 7216747374720, 123128587713600, 2223511629522624, 42370586275466880, 849664985938704000, 17886165587251839360, 394366490810199895680, 9088843342633833461760
Offset: 3

Views

Author

Len Smiley, Dec 12 2001

Keywords

Examples

			a(4)=8: ring must have 3 of the four, fourth in middle. Two ways for the three to hold hands.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Links

Crossrefs

Cf. A066166 (original version).

Programs

  • Mathematica
    max = 20; f[x_] := Exp[-x*Log[1 - x] - x^2] - 1; Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!, 3] (* Jean-François Alcover, Oct 13 2011, after g.f. *)
  • Maxima
    a(n):=n!*sum(sum(binomial(k,j)*j!/(n-2*k+j)!*stirling1(n-2*k+j,j)*(-1)^(n-k-j),j,0,k)/k!,k,1,floor(n/2)); /* Vladimir Kruchinin, Sep 07 2010 */

Formula

E.g.f.: exp(-x*log(1-x)-x^2)-1.
a(n) = n!*sum(sum(binomial(k,j)*j!/(n-2*k+j)!*Stirling1(n-2*k+j,j)*(-1)^(n-k-j),j,0,k)/k!,k,1,floor(n/2)), n>2. - Vladimir Kruchinin, Sep 07 2010
a(n) ~ exp(-1) * n!. - Vaclav Kotesovec, Jun 04 2022
Previous Showing 11-20 of 24 results. Next

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