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.

A265954 Coefficients in asymptotic expansion of sequence A265950.

Original entry on oeis.org

1, 1, 2, 10, 56, 394, 3332, 32782, 368072, 4651666, 65440748, 1015205974, 17225810768, 317432262298, 6313880504564, 134828046043486, 3076458785723864, 74696205255843490, 1922729345267645180, 52297599798809376358, 1498690940537194229600
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 19 2015

Keywords

Examples

			A265950(n) / n! ~ 1 + 1/n + 2/n^2 + 10/n^3 + 56/n^4 + 394/n^5 + 3332/n^6 + ...

Links

Crossrefs

Cf. A077365, A256125, A265950.

Formula

a(k) ~ k! / (4 * (log(2))^(k+1)).

A007837 Number of partitions of n-set with distinct block sizes.

Original entry on oeis.org

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847
Offset: 0

Views

Author

Arnold Knopfmacher

Keywords

Comments

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

Examples

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Links

Crossrefs

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011
  • Mathematica
    nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)
  • PARI
    {my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

Formula

E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

Extensions

More terms from Christian G. Bower
a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A077365 Sum of products of factorials of parts in all partitions of n.

Original entry on oeis.org

1, 1, 3, 9, 37, 169, 981, 6429, 49669, 430861, 4208925, 45345165, 536229373, 6884917597, 95473049469, 1420609412637, 22580588347741, 381713065286173, 6837950790434781, 129378941557961565, 2578133190722896861, 53965646957320869469, 1183822028149936497501
Offset: 0

Views

Author

Vladeta Jovovic, Nov 30 2002

Keywords

Comments

Row sums of arrays A069123 and A134133. Row sums of triangle A134134.

Examples

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products of factorials of parts are 24,6,4,2,1 and their sum is a(4) = 37.
1 + x + 3 x^2 + 9 x^3 + 37 x^4 + 169 x^5 + 981 x^6 + 6429 x^7 + 49669 x^8 + ...

Links

Crossrefs

Cf. A006906, A074141, A256125, A265950.
Cf. A069123, A134133, A134134.
Cf. A051296 (with compositions instead of partitions).

Programs

  • Maple
    b:= proc(n, i, j) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
      `if`(i>n, 0, j^i*b(n-i, i, j+1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 03 2013
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, i)*i!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016
  • Mathematica
    Table[Plus @@ Map[Times @@ (#!) &, IntegerPartitions[n]], {n, 0, 20}] (* Olivier Gérard, Oct 22 2011 *)
a[ n_] := If[ n < 0, 0, Plus @@ Times @@@ (IntegerPartitions[ n] !)] (* Michael Somos, Feb 09 2012 *)
nmax=20; CoefficientList[Series[Product[1/(1-k!*x^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)
b[n_, i_, j_] := b[n, i, j] = If[n==0, 1, If[i<1, 0, b[n, i-1, j] + If[i>n, 0, j^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)
  • PARI
    N=66; q='q+O('q^N);
gf= 1/prod(n=1,N, (1-n!*q^n) );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

Formula

G.f.: 1/Product_{m>0} (1-m!*x^m).
Recurrence: a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(k/d).
a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 67/n^4 + 457/n^5 + 3734/n^6 + 35741/n^7 + 392875/n^8 + 4886114/n^9 + 67924417/n^10), for coefficients see A256125. - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

Extensions

Unnecessarily complicated mma code deleted by N. J. A. Sloane, Sep 21 2009

A292279 Expansion of 1/Product_{k>=1} (1 + k!*x^k).

Original entry on oeis.org

1, -1, -1, -5, -15, -93, -551, -4129, -33607, -312929, -3179343, -35602881, -432201743, -5678740945, -80142780751, -1210609725905, -19481112885231, -332836223507793, -6016678424942063, -114746996449871761, -2302527084416470255, -48495552665272893329
Offset: 0

Views

Author

Seiichi Manyama, Sep 13 2017

Keywords

Links

Crossrefs

Cf. A077365, A265950.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[1/(1 + k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2017 *)
  • PARI
    {a(n) = polcoeff(1/prod(k=1, n, 1+k!*x^k+x*O(x^n)), n)}

Formula

Convolution inverse of A265950.
a(n) ~ -n! * (1 - 1/n - 1/n^2 - 6/n^3 - 31/n^4 - 219/n^5 - 1932/n^6 - 19945/n^7 - 234837/n^8 - 3104108/n^9 - 45495817/n^10). - Vaclav Kotesovec, Sep 14 2017
G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*(j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

A292280 Expansion of Product_{k>=1} (1 - k!*x^k).

Original entry on oeis.org

1, -1, -2, -4, -18, -84, -564, -3984, -33504, -307728, -3156192, -35254080, -429350400, -5641133760, -79720588800, -1204747741440, -19400679325440, -331599765565440, -5996988417784320, -114408970262922240, -2296442484579686400, -48379417944213196800
Offset: 0

Views

Author

Seiichi Manyama, Sep 13 2017

Keywords

Links

Crossrefs

Cf. A077365, A265950.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1 - k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 13 2017 *)
  • PARI
    {a(n) = polcoeff(prod(k=1, n, 1-k!*x^k+x*O(x^n)), n)}

Formula

Convolution inverse of A077365.
a(n) ~ -n! * (1 - 1/n - 2/n^2 - 6/n^3 - 32/n^4 - 222/n^5 - 1916/n^6 - 19650/n^7 - 231200/n^8 - 3058566/n^9 - 44883428/n^10). - Vaclav Kotesovec, Sep 14 2017
G.f.: exp(-Sum_{k>=1} Sum_{j>=1} (j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

A321520 Expansion of Product_{k>=1} (1 + (k - 1)!*x^k).

Original entry on oeis.org

1, 1, 1, 3, 8, 32, 152, 882, 5964, 46644, 411564, 4056912, 44097072, 524234448, 6761911968, 94055452128, 1403047948320, 22342552398720, 378256278306240, 6783950610708480, 128480976137122560, 2562250754919421440, 53668564630447910400
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 12 2018

Keywords

Links

Crossrefs

Cf. A107107, A265950.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 05 2023
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + (k - 1)! x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d ((d - 1)!)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 22}]

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*((d - 1)!)^(k/d) ) * x^k/k).
a(n) ~ (n-1)! * (1 + 1/n + 2/n^2 + 7/n^3 + 34/n^4 + 203/n^5 + 1454/n^6 + 12321/n^7 + 121326/n^8 + 1364947/n^9 + 17301550/n^10 + ...). - Vaclav Kotesovec, Nov 13 2018

A158615 Expansion of Sum_{n>0} n*n!*x^n/(1-n!*x^n).

Original entry on oeis.org

1, 5, 19, 105, 601, 4445, 35281, 324897, 3266569, 36360065, 439084801, 5751188913, 80951270401, 1220673888257, 19615124183329, 334777645154817, 6046686277632001, 115243914079782593, 2311256907767808001
Offset: 1

Views

Author

Vladeta Jovovic, Mar 22 2009

Keywords

Comments

a(n) = Sum_{d|n} d*d!^(n/d).

Links

Crossrefs

Cf. A077365, A265950.

Programs

  • Maple
    nmax := 40: gf := add( taylor( n*n!*x^n/(1-n!*x^n),x=0,nmax+1),n=1..nmax ) : coeffs(convert(gf,polynom)) ; # R. J. Mathar, Mar 30 2009
  • Mathematica
    nmax=20; Rest[CoefficientList[Series[Sum[k*k!*x^k/(1-k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 19 2015 *)

Formula

a(n) ~ n * n!. - Vaclav Kotesovec, Dec 19 2015

Extensions

More terms from R. J. Mathar, Mar 30 2009

A292318 Expansion of Product_{k>=1} ((1 + k!*x^k)/(1 - k!*x^k)).

Original entry on oeis.org

1, 2, 6, 22, 90, 434, 2442, 15874, 118722, 1009586, 9640866, 102243682, 1191949122, 15141785570, 208068223458, 3073613823778, 48554040330210, 816547584905186, 14562214993474914, 274463503469613538, 5450631032885614050, 113749623991878727394
Offset: 0

Views

Author

Seiichi Manyama, Sep 14 2017

Keywords

Crossrefs

Cf. A077365, A265950, A292319.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Product[(1 + k!*x^k)/(1 - k!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 15 2017 *)

Formula

Convolution of A077365 and A265950.
a(n) ~ 2 * n! * (1 + 2/n + 6/n^2 + 28/n^3 + 162/n^4 + 1134/n^5 + 9368/n^6 + 89502/n^7 + 974338/n^8 + 11948360/n^9 + 163462518/n^10). - Vaclav Kotesovec, Sep 15 2017

A318247 a(n) = [x^n] Product_{k>=1} (1 + n!*x^k).

Original entry on oeis.org

1, 1, 2, 42, 600, 28920, 374285520, 128100273840, 131101518683520, 143354704247556480, 173401404266683545849388800, 2538767479410416957720411116800, 105287752487031026606448840363801600, 4510685217145833106538730603088118860800, 288804138719404983322786510403231912442931200
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 22 2018

Keywords

Links

Crossrefs

Cf. A077365, A265950, A292306, A292414, A318246.

Programs

  • Mathematica
    nmax = 15; Table[SeriesCoefficient[Product[(1+n!*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Showing 1-9 of 9 results.

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