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 24 results. Next

A351737 Expansion of e.g.f. exp( x * (exp(3 * x) - 1) ).

Original entry on oeis.org

1, 0, 6, 27, 216, 2025, 21708, 260253, 3460320, 50395041, 795324420, 13495904829, 244747554912, 4718754452529, 96285948702804, 2071265238290565, 46815054101658432, 1108489016781839169, 27424412680091114628, 707277138662880504045, 18974871706141125008640
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Links

Crossrefs

Cf. A052506, A351736.
Cf. A351734, A351735.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(3*x)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-k)*stirling(n-k, k, 2)/(n-k)!);
  • PARI
    a(n) = sum(k=0, n, (3*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(3*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 3^(n-k) * Stirling2(n-k,k)/(n-k)!.
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (3*k-1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} x^k / (1 - (3*k-1)*x)^(k+1). (End)

A351736 Expansion of e.g.f. exp( x * (exp(2 * x) - 1) ).

Original entry on oeis.org

1, 0, 4, 12, 80, 560, 4512, 40768, 407808, 4453632, 52605440, 667234304, 9032423424, 129822564352, 1972450443264, 31559866736640, 530043925495808, 9317136303718400, 170976603113127936, 3268020569256755200, 64928967058257346560, 1338431135849666052096
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Links

Crossrefs

Cf. A052506, A351737.
Cf. A053491, A351733.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(2*x)-1))))
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-k)*stirling(n-k, k, 2)/(n-k)!);
  • PARI
    a(n) = sum(k=0, n, (2*k-1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(2*k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * Stirling2(n-k,k)/(n-k)!.
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (2*k-1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} x^k / (1 - (2*k-1)*x)^(k+1). (End)

A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 06 2014

Keywords

Links

Crossrefs

Column k=2 of A292892.
Cf. A000110, A052506, A216507, A060311.
Cf. A216688, A216689, A080108, A000248.

Programs

  • Mathematica
    CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!
  • PARI
    x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017
  • PARI
    a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

Formula

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

A349560 E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * x.

Original entry on oeis.org

1, 0, 2, 3, 40, 245, 2976, 35287, 524560, 8790777, 165530800, 3493679651, 80812685064, 2049413147509, 56294089065592, 1668771901644135, 53057068616526496, 1801519375618579313, 65063987978980974048, 2490449984485154892235, 100716775979173952155480
Offset: 0

Views

Author

Seiichi Manyama, Nov 22 2021

Keywords

Crossrefs

Cf. A052506, A349557.
Cf. A356785, A356788, A356789.

Programs

  • Maple
    a:= n-> n!*coeff(series(RootOf(A=exp(x*exp(x*A)-x), A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 22 2021
  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = Exp[(E^(x*A[x]) - 1)*x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)
  • PARI
    my(A=1,n=22); for(i=1, n, A=exp((exp(x*A)-1)*(x+x*O(x^n)))); Vec(serlaplace(A))
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Aug 27 2022

Formula

a(n) ~ sqrt(s*(1 - r^2*s/(1 + r*s))) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.4599551063707173872728335298048828687860291021728... is the root of the equation r - LambertW(1/r) - 2*log(r) = 1/LambertW(1/r) and s = LambertW(1/r)/r = 1.938208283387405345404104769972407921289092368509... - Vaclav Kotesovec, Nov 22 2021
a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - Seiichi Manyama, Aug 27 2022

A354001 Expansion of e.g.f. exp(x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 616, 5124, 29520, 138765, 942700, 9369646, 91711984, 782281955, 6539493520, 62576274440, 693828386976, 7968383514969, 89851862221140, 1023732374445970, 12384993316732960, 160496534000858671, 2163244034675904664, 29653387436468336300
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Links

Crossrefs

Cf. A052506, A354000.
Cf. A351493, A353999.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x^3/6 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 07 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(exp(x)-1))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=4, i, j/(j-3)!*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=4..n} k/(k-3)! * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A292892 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 6, 16, 52, 1, 0, 0, 0, 12, 65, 203, 1, 0, 0, 0, 24, 20, 336, 877, 1, 0, 0, 0, 0, 60, 390, 1897, 4140, 1, 0, 0, 0, 0, 120, 120, 2562, 11824, 21147, 1, 0, 0, 0, 0, 0, 360, 210, 11816, 80145, 115975, 1, 0, 0, 0, 0, 0, 720, 840, 20496, 105912, 586000, 678570
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,  1,   1, ...
   1,  0,  0,  0,   0, ...
   2,  2,  0,  0,   0, ...
   5,  3,  6,  0,   0, ...
  15, 16, 12, 24,   0, ...
  52, 65, 20, 60, 120, ...

Links

Crossrefs

Columns k=0..3 give A000110, A052506, A240989, A292891.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
Cf. A292894, A355607.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

A292893 Expansion of e.g.f. exp(x * (1 - exp(x))).

Original entry on oeis.org

1, 0, -2, -3, 8, 55, 84, -637, -4992, -10593, 92060, 1012099, 3642000, -18733585, -354606084, -2157876645, 2003383424, 175455790399, 1766183783868, 5436448194707, -96997103373360, -1770215099996721, -13073420293290148, 22275369715313131
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2017

Keywords

Links

Crossrefs

Column k=1 of A292894.
Cf. A000587, A052506, A080108.

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x*(1-exp(x)))))
  • PARI
    a(n) = n!*sum(k=0, n\2, (-1)^k*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Jul 09 2022
  • 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/(j-1)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022
  • PARI
    a(n) = sum(k=0, n, (-1)^k*(k+1)^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Aug 29 2022
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * Stirling2(n-k,k)/(n-k)!.
a(0) = 1; a(n) = -(n-1)! * Sum_{k=2..n} k/(k-1)! * a(n-k)/(n-k)!. (End)
From Seiichi Manyama, Aug 29 2022: (Start)
a(n) = Sum_{k=0..n} (-1)^k * (k+1)^(n-k) * binomial(n,k).
G.f.: Sum_{k>=0} (-x)^k / (1 - (k+1)*x)^(k+1). (End)

A356806 a(n) = Sum_{k=0..n} (k*n-1)^(n-k) * binomial(n,k).

Original entry on oeis.org

1, 0, 4, 27, 448, 10625, 344736, 14437213, 753991680, 47974773393, 3650824000000, 326917384798301, 33956137832546304, 4041303651931462969, 545552768347831566336, 82828479894303251953125, 14040577418634835164921856, 2640293357854435329683551265
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Links

Crossrefs

Cf. A052506, A351736, A351737.
Cf. A356811, A356814, A356817.

Programs

  • PARI
    a(n) = sum(k=0, n, (k*n-1)^(n-k)*binomial(n, k));
  • PARI
    a(n) = n!*sum(k=0, n\2, n^(n-k)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * [x^n] exp( x * (exp(n * x) - 1) ).
a(n) = n! * Sum_{k=0..floor(n/2)} n^(n-k) * Stirling2(n-k,k)/(n-k)!.
a(n) = [x^n] Sum_{k>=0} x^k / (1 - (n*k-1)*x)^(k+1).

A354000 Expansion of e.g.f. exp(x^2/2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 105, 651, 2968, 18936, 152505, 1164295, 9109056, 80012868, 756041377, 7387199925, 75535791360, 816560002576, 9254683835073, 109135702334619, 1338613513677280, 17079079303721820, 226148006163689841, 3100114305453613393, 43935964285680790368
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Links

Crossrefs

Cf. A052506, A354001.
Cf. A353998, A351492.

Programs

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

Formula

a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(2^k * (n-2*k)!).

A356797 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^2.

Original entry on oeis.org

1, 0, 2, 3, 64, 305, 6936, 64897, 1645008, 24290289, 692240680, 14243244521, 456748635432, 12105737521033, 435619742434800, 14112089558682585, 567134312211275296, 21653262317886286817, 966207399513747354072, 42358800314758614030505
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2022

Keywords

Links

Crossrefs

Cf. A052506, A355843, A356798.
Cf. A356788.

Programs

  • Mathematica
    m = 20; (* number of terms *)
CoefficientList[Exp[-(1/2)*LambertW[-2*(Exp[x]-1)*x]] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 11 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*(1-exp(x)))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*(1-exp(x)))/(2*x*(1-exp(x))))^(1/2)))

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(2 * x * (1 - exp(x)))/2 ).
E.g.f.: A(x) = ( LambertW(2 * x * (1 - exp(x)))/(2 * x * (1 - exp(x))) )^(1/2).
Showing 1-10 of 24 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.