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 11 results. Next

A174661 E.g.f.: Product_{n>=1} 1/(1 - x^n/n!)^n.

Original entry on oeis.org

1, 1, 4, 15, 82, 475, 3456, 26719, 239996, 2313609, 24846640, 285861301, 3586817928, 47988744115, 690525294018, 10547453864445, 171595180564816, 2949836193259105, 53630566660122696, 1025856767305899229, 20638503314068334480
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2010

Keywords

Comments

Compare to the e.g.f. of A032315: Product_{n>=1} (1 + x^n/n!)^n.

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 15*x^3/3! + 82*x^4/4! + ...;
A(x) = 1/((1-x)*(1-x^2/2!)^2*(1-x^3/3!)^3*(1-x^4/4!)^4*(1-x^5/5!)^5* ...).

Links

Crossrefs

Cf. variant: A032315.

Programs

  • PARI
    {a(n)=n!*polcoeff(prod(k=1,n,1/(1-x^k/k!+x*O(x^n))^k),n)}

Formula

a(n) ~ c * n!, where c = product_{k>=2} 1/(1-1/k!)^k = 8.6304199482678945455168174204973507297310235756... . - Vaclav Kotesovec, Nov 03 2014
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} j*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Sep 12 2018

A341506 E.g.f.: Product_{i>=1, j>=1} (1 + x^(i*j) / (i*j)!).

Original entry on oeis.org

1, 1, 2, 8, 17, 87, 366, 1514, 8770, 45585, 267586, 1612624, 11914416, 73215391, 522906754, 4364545708, 33150679697, 263662491935, 2151338992440, 20815916251604, 178593028936507, 1714283809331191, 15531842607259512, 158682350653110712, 1667852117293837230
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 13 2021

Keywords

Links

Crossrefs

Cf. A000005, A007837, A032315, A107742, A318694, A340903, A341505, A341876.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Product[(1 + x^k/k!)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, -(n - 1)! Sum[Sum[d DivisorSigma[0, d]/(-d!)^(k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

Formula

E.g.f.: Product_{k>=1} (1 + x^k / k!)^sigma_0(k).

A294494 E.g.f.: 1/Product_{k>0} (1+x^k/k!)^k.

Original entry on oeis.org

1, -1, 0, -3, 26, -75, 324, -3535, 30988, -242025, 2245820, -26847381, 339741984, -4205748547, 57094691822, -883946426805, 14358210544304, -239959114870689, 4286519116236900, -82194727064059645, 1650577120959962440, -34495065863164195611
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A292308.
Cf. A032315, A174661, A294495.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/Product[(1+x^k/k!)^k,{k,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 12 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, (1+x^k/k!)^k)))

A294495 E.g.f.: Product_{k>0} (1-x^k/k!)^k.

Original entry on oeis.org

1, -1, -2, 3, 14, 45, -156, -1225, -3396, -105, 226760, 1175229, 4084200, -35683219, -585896962, -3512021955, -14398868176, 198247498911, 3131185307832, 29821940715413, 122481857683680, -1187008881417051, -31616420134183522, -419944298964036771
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A185895.
Cf. A032315, A174661, A294494.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1-x^k/k!)^k)))

A319175 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k/k!)^n.

Original entry on oeis.org

1, 1, 4, 36, 416, 6000, 106542, 2242093, 54399424, 1495318752, 45938780750, 1559858659359, 58007497143180, 2344682328265823, 102352889947823998, 4798930456964580045, 240518006611511552896, 12832137350594892322464, 726108032647676403262710, 43434461707962856186584307
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 12 2018

Keywords

Crossrefs

Cf. A007837, A032312, A032314, A032315, A319174, A319177.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[(1 + x^k/k!)^n, {k, 1, n}], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[(-1)^(k + 1) x^(j k)/(k (j!)^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 19}]

Formula

a(n) = n! * [x^n] exp(n*Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)).

A371309 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(k^2).

Original entry on oeis.org

1, 1, 4, 21, 88, 645, 4386, 33061, 296808, 2674377, 26757190, 285294801, 3199971336, 38104015417, 476816609192, 6261875218215, 85736036949856, 1227373767823617, 18255004438680558, 281702101344334993, 4509097743074878200, 74606738071821274641, 1275212550939684334384
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of squares.

Links

Crossrefs

Cf. A000290, A007837, A027998, A032315, A032316.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A371482 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(2^k).

Original entry on oeis.org

1, 2, 6, 32, 164, 1032, 7728, 59376, 522600, 4946768, 49680656, 540031296, 6195155744, 75183755584, 961596510272, 12909563309952, 181305865742240, 2657525771641664, 40594443765953472, 643987597483557888, 10601112599585001984, 180727870834447607808, 3185418524574895953152
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of powers of 2.

Links

Crossrefs

Cf. A000079, A007837, A032315, A032316, A102866, A199163.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^(2^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A371308 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^prime(k).

Original entry on oeis.org

1, 2, 5, 23, 101, 511, 3300, 20499, 147249, 1158047, 9284124, 82250155, 762408746, 7406758725, 75928931645, 815389826454, 9127145085135, 106002459387831, 1287304713397098, 16132127163478581, 209381715443456410, 2814011969429674997, 38957100435462040565
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 24 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of primes.

Links

Crossrefs

Cf. A000040, A007837, A032315, A032316, A061152.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A371480 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(k-1).

Original entry on oeis.org

1, 0, 1, 2, 3, 24, 70, 300, 1365, 10136, 48636, 253560, 2069430, 13716780, 107972579, 695118606, 6155201325, 53001340112, 483401450344, 4129645998960, 36957700537146, 398523920633940, 3954961713704673, 42541112692428746, 430044528651402148
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of 0,1,2,3,4,...

Links

Crossrefs

Cf. A007837, A032315, A032316, A052812.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Product[(1 + x^k/k!)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A371481 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 3, 15, 52, 335, 2031, 12880, 102614, 802443, 6956995, 64721756, 633754320, 6551212057, 71375943289, 820250162880, 9747543483676, 121775559807881, 1580353806494781, 21246545374234378, 296590230821338520, 4280692741624646151, 63852747607056438283
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2024

Keywords

Comments

"EGJ" (unordered, element, labeled) transform of triangular numbers.

Links

Crossrefs

Cf. A000217, A007837, A028377, A032315, A032316, A371309.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k/k!)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-10 of 11 results. Next

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

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