A247551 -id:A247551 - cp's OEIS frontend

A076900 Expansion of e.g.f.: 1/Product_{m>0} (1-x^m/(m-1)!).

1, 1, 4, 15, 88, 505, 4056, 31549, 311816, 3083049, 36343720, 431215741, 5937234348, 82236865165, 1291252453050, 20477737537755, 361495828272496, 6449450737736065, 126566562342343176, 2509520619696338269, 54179963857121953460, 1182248224137860933781

Offset: 0

Vladeta Jovovic, Nov 26 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A005651, A032299.

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)*binomial(n, i)*i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + If[i > n, 0, b[n-i, i] Binomial[n, i] i]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Nov 03 2020, after Alois P. Heinz *)

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*((j - 1)!)^k)). - Ilya Gutkovskiy, Sep 13 2018

a(n) ~ c * n * n!, where c = A247551/2. - Vaclav Kotesovec, Sep 13 2018

A266518 Number of ordered partitions of a 2n-set with nondecreasing block sizes and maximal block size equal to n.

Original entry on oeis.org

1, 2, 18, 200, 3290, 61992, 1480248, 39402792, 1229123610, 42349478600, 1640551617848, 69364811821032, 3222214209737432, 161656803984848200, 8772238289222220600, 509677254444910662000, 31677425399312755814970, 2092539622373193784503240

Offset: 0

Alois P. Heinz, Dec 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A262071, A327801, A328156.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
    end:
a:= n-> `if`(n=0, 1, b(2*n, n)-b(2*n, n-1)):
seq(a(n), n=0..20);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Binomial[n, i]*b[n-i, i]]]]; a[n_] := If[n==0, 1, b[2n, n] - b[2n, n-1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

a(n) = (2n)! * [x^n] Product_{i=1..n} (i-1)!/(i!-x^i).
a(n) = A262071(2n,n).
a(n) ~ c * 2^(2*n+1/2) * n^n / exp(n), where c = A247551 = 2.529477472079152648... . - Vaclav Kotesovec, Jan 02 2016
a(n) = A327801(2n,n). - Alois P. Heinz, Sep 26 2019

A292796 Number of sets of nonempty words with a total of n letters over n-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 3, 13, 60, 326, 2065, 14508, 116845, 1039459, 10339365, 112376487, 1339665295, 17256611005, 240193792120, 3578746993871, 56986570945387, 963868021665359, 17281651020455445, 327058650473873893, 6519981694119182165, 136489249161324882063

Offset: 0

Alois P. Heinz, Sep 23 2017

nonn

			a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 3: {aa}, {ab}, {ba}.
a(3) = 13: {aaa}, {aab}, {aba}, {baa}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}, {aa,a}, {ab,a}, {ba,a}.

Alois P. Heinz, Table of n, a(n) for n = 0..450

Main diagonal of A292795.
Row sums of A319498.
Cf. A226873, A292713.

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
    end:
a:= n-> h(n$3):
seq(a(n), n=0..30);

b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
g[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]];
h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[g[i, k], j], {j, 0, n/i}]]];
a[n_] := h[n, n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

a(n) = [x^n] Product_{j=1..n} (1+x^j)^A226873(j,n).
a(n) = A292795(n,n).
a(n) ~ c * n!, where c = A247551 = 2.529477472079152648... - Vaclav Kotesovec, Sep 28 2017

A335643 Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k!).

Original entry on oeis.org

1, 1, 3, 12, 71, 462, 3890, 35133, 381583, 4411870, 58623990, 826335675, 12990713482, 216027857567, 3925135187017, 75217607162053, 1552186877466271, 33678081631793270, 778592124168964502, 18867293553102673343, 483291402186818709310, 12937553749692179771301, 363847628395565829224327

Offset: 0

Seiichi Manyama, Oct 03 2020

nonn

Cf. A005651, A335627, A335636, A335642, A336046.

max = 22; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Tan[x]^k/k!), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-tan(x)^k/k!)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, tan(x)^(i*j)/(i*j!^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} tan(x)^(i*j)/(i*(j!)^i) ).

a(n) ~ A247551 * 2^(2*n+1) * n! / Pi^(n+1). - Vaclav Kotesovec, Oct 04 2020

A327827 Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating 1.

Original entry on oeis.org

0, 1, 2, 9, 40, 235, 1476, 11214, 91848, 859527, 8710300, 97675138, 1179954612, 15490520786, 217602374458, 3280028076615, 52571985879600, 895913825750191, 16140560853800556, 307048409240931810, 6143666813617775100, 129096480664676773542, 2840750997343361802150

Offset: 0

Alois P. Heinz, Sep 26 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Column k=1 of A327801.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1, `if`(i=k, 0, k))+
     `if`(i=k, 0, b(n-i, i, k)*binomial(n, i))))
    end:
a:= n-> b(n, 1, 0)-b(n, 1$2):
seq(a(n), n=0..23);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i], k]/i!]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
a[n_] := T[n, 1];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

a(n) ~ c * n!, where c = A247551 = 2.5294774720791526481801161542539542411787... - Vaclav Kotesovec, Sep 28 2019

A370413 Decimal expansion of Product_{k>=2} 1 / (1 - 1/k!!).

Original entry on oeis.org

3, 8, 0, 3, 0, 8, 9, 1, 5, 2, 3, 7, 5, 6, 2, 0, 1, 4, 0, 5, 8, 2, 5, 7, 4, 4, 9, 6, 8, 5, 4, 9, 9, 0, 0, 3, 8, 4, 5, 9, 0, 7, 4, 4, 4, 2, 7, 1, 3, 8, 2, 2, 1, 7, 1, 6, 5, 5, 9, 2, 8, 3, 2, 4, 5, 3, 8, 3, 2, 4, 2, 4, 6, 0, 3, 6, 5, 2, 1, 5, 6, 9, 6, 6, 3, 7, 7, 3, 0, 8, 9, 8, 8, 6, 7, 1, 3, 6, 7, 3

Offset: 1

Ilya Gutkovskiy, Mar 30 2024

			3.803089152375620140582574496854990038459...

Cf. A006882, A247551, A369994.

A327828 Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating 2.

Original entry on oeis.org

0, 0, 1, 3, 18, 100, 705, 5166, 44856, 413316, 4297635, 47906650, 586050828, 7669704978, 108433645502, 1632017808435, 26240224612920, 446861879976600, 8063224431751719, 153335328111105282, 3070484092409318100, 64508501542986638550, 1420061287311444508962

Offset: 0

Alois P. Heinz, Sep 26 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Column k=2 of A327801.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1, `if`(i=k, 0, k))+
     `if`(i=k, 0, b(n-i, i, k)*binomial(n, i))))
    end:
a:= n-> b(n, 1, 0)-b(n, 1, 2):
seq(a(n), n=0..23);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i > n, 0, b[n, i + 1, If[i == k, 0, k]] + If[i == k, 0, b[n - i, i, k] Binomial[n, i]]]];
a[n_] := b[n, 1, 0] - b[n, 1, 2];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

a(n) ~ c * n!, where c = A247551/2 = 1.26473873603957632409005807712697712... - Vaclav Kotesovec, Sep 28 2019

A330649 E.g.f.: Product_{k>=1} 1 / (1 - x^k/(k!*(1 - x)^k)).

Original entry on oeis.org

1, 1, 5, 34, 299, 3226, 41202, 607545, 10153831, 189628750, 3913009178, 88406043991, 2170372901534, 57531498837515, 1637713270797411, 49830222530823615, 1613950394999111903, 55444724259894089718, 2013760368429942861810, 77105255895256112519259

Offset: 0

Ilya Gutkovskiy, Feb 13 2020

nonn

Cf. A005651, A084358, A218482, A320566, A332024.

nmax = 19; CoefficientList[Series[Product[1/(1 - x^k/(k! (1 - x)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Total[Apply[Multinomial, IntegerPartitions[k], {1}]] n!/k!, {k, 0, n}], {n, 0, 19}]

seq(n)={Vec(serlaplace(prod(k=1, n, 1 / (1 - x^k/(k!*(1 - x)^k)) + O(x*x^n))))} \\ Andrew Howroyd, Feb 13 2020

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * A005651(k) * n! / k!.

a(n) ~ c * 2^(n-1) * n!, where c = A247551 = 2.52947747207915264818... - Vaclav Kotesovec, Feb 16 2020

A370465 Decimal expansion of Product_{k>=2} 1 / (1 - 1/(2*k-1)!!).

Original entry on oeis.org

1, 6, 2, 4, 4, 8, 4, 1, 5, 2, 3, 4, 5, 4, 5, 5, 8, 4, 2, 4, 5, 0, 8, 8, 2, 1, 6, 9, 9, 7, 9, 7, 2, 3, 7, 4, 9, 9, 0, 1, 9, 4, 5, 7, 1, 0, 0, 5, 3, 5, 6, 6, 7, 3, 9, 3, 3, 3, 0, 9, 9, 1, 4, 8, 0, 8, 7, 1, 6, 2, 2, 7, 9, 1, 4, 4, 9, 7, 0, 6, 6, 5, 5, 3, 7, 8, 7, 4, 0, 8, 3, 5, 2, 3, 7, 6, 5, 8, 8, 9

Offset: 1

Ilya Gutkovskiy, Mar 30 2024

			1.6244841523454558424508821699797237499...

Cf. A001147, A247551, A370424.

A371312 Expansion of e.g.f. Product_{k>=1} 1 / (1 - x^k/k!)^2.

Original entry on oeis.org

1, 2, 8, 38, 228, 1562, 12386, 109286, 1073988, 11545994, 135393438, 1714890806, 23380747506, 341014477390, 5303722839850, 87582446980418, 1531259993710468, 28254163132485930, 548854481037814382, 11196310379931318758, 239346426732701009838, 5350768890908294837294

Offset: 0

Ilya Gutkovskiy, Mar 24 2024

nonn

Exponential self-convolution of A005651.

Cf. A005651, A032312.

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k/k!)^2, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

a(n) = Sum_{k=0..n} binomial(n,k) * A005651(k) * A005651(n-k).

a(n) ~ A247551^2 * n! * n. - Vaclav Kotesovec, Mar 24 2024

cp's OEIS Frontend

A076900 Expansion of e.g.f.: 1/Product_{m>0} (1-x^m/(m-1)!).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A266518 Number of ordered partitions of a 2n-set with nondecreasing block sizes and maximal block size equal to n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A292796 Number of sets of nonempty words with a total of n letters over n-ary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A335643 Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A327827 Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A370413 Decimal expansion of Product_{k>=2} 1 / (1 - 1/k!!).

Views

Author

Keywords

Examples

Crossrefs

A327828 Sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330649 E.g.f.: Product_{k>=1} 1 / (1 - x^k/(k!*(1 - x)^k)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A370465 Decimal expansion of Product_{k>=2} 1 / (1 - 1/(2*k-1)!!).

Views

Author