A320566 -id:A320566 - cp's OEIS frontend

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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

Offset: 1

Vaclav Kotesovec, Sep 19 2014

			2.5294774720791526481801161542539542411787023948459973375849349825...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 269.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R2.

Cf. A005651, A035341, A072605, A140585, A183239, A209668, A261600, A282529, A292713, A320566, A327827, A335643.

evalf(1/product(1-1/k!,k=2..infinity),100); # Vaclav Kotesovec, Sep 19 2014

digits = 105;
RealDigits[NProduct[1/(1-1/k!), {k, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> digits], 10, digits][[1]] (* Jean-François Alcover, Nov 02 2020 *)

default(realprecision,150); 1/prodinf(k=2,1 - 1/k!) \\ Vaclav Kotesovec, Sep 21 2014

Product_{k>=2} 1/(1-1/k!).
Equals lim_{n -> oo} A005651(n) / n!.
Equals 1/A282529. - Amiram Eldar, Sep 15 2023

A327801 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 47, 40, 18, 4, 1, 246, 235, 100, 30, 5, 1, 1602, 1476, 705, 200, 45, 6, 1, 11481, 11214, 5166, 1645, 350, 63, 7, 1, 95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1, 871030, 859527, 413316, 134568, 30996, 5922, 840, 108, 9, 1

Offset: 0

Alois P. Heinz, Sep 25 2019

Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.

			Triangle T(n,k) begins:
      1;
      1,     1;
      3,     2,     1;
     10,     9,     3,     1;
     47,    40,    18,     4,    1;
    246,   235,   100,    30,    5,   1;
   1602,  1476,   705,   200,   45,   6,  1;
  11481, 11214,  5166,  1645,  350,  63,  7, 1;
  95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Columns k=0-2 give: A005651, A327827, A327828.
Row sums give A320566.
T(2n,n) gives A266518.
T(n,n-1) gives A001477.
T(n+1,n-1) gives A045943.
Cf. A327869.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> k=0 or k in x, partition(n))):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
b:= proc(n, i, k) option remember; `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!))
    end:
T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..10);

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]]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, from 2nd Maple program *)

A320567 Expansion of e.g.f. exp(x) * Product_{k>=1} (1 + x^k/k!).

Original entry on oeis.org

1, 2, 4, 11, 32, 97, 355, 1423, 5696, 23141, 108149, 559693, 2913971, 14806365, 75692999, 432849976, 2780749376, 18237870285, 115493756737, 708062095921, 4354275076517, 29539724932771, 227955214198529, 1836106089485736, 14279737884301139, 105409744347318897

Offset: 0

Ilya Gutkovskiy, Oct 15 2018

nonn

Binomial transform of A007837.

Alois P. Heinz, Table of n, a(n) for n = 0..692
N. J. A. Sloane, Transforms

Cf. A007837, A320566.

seq(coeff(series(factorial(n)*exp(x)*mul(1+x^k/factorial(k),k=1..n),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 15 2018
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
    end:
a:= n-> add(b(n-i)*binomial(n, i), i=0..n):
seq(a(n), n=0..27);  # Alois P. Heinz, Sep 27 2019

nmax = 25; CoefficientList[Series[Exp[x] Product[(1 + x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 25; CoefficientList[Series[Exp[x + Sum[Sum[(-1)^(k + 1) x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(x + Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)).

a(n) = Sum_{k=0..n} binomial(n,k)*A007837(k).

A327870 Row sums of A327869.

Original entry on oeis.org

1, 2, 2, 11, 14, 47, 305, 611, 2070, 8831, 84077, 204371, 944333, 3850407, 23991739, 297448526, 927586630, 4775902567, 24534836837, 141681919871, 1080484165089, 18553632475991, 66762080435239, 415657332495526, 2298883231736949, 15799818116227747

Offset: 0

Alois P. Heinz, Sep 28 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Row sums of A327869.

Cf. A320566.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 b(n$2, 0)*(n+1) -add(b(n$2, k), k=1..n):
seq(a(n), n=0..28);

b[n_, i_, k_] := b[n, i, k] = If[i*(i + 1)/2 < n, 0,
     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 - 1], k]*Binomial[n, i]]]];
a[n_] := b[n, n, 0]*(n + 1) - Sum[b[n, n, k], {k, 1, n}];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Mar 28 2022, after Alois P. Heinz *)

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

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A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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A327801 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A320567 Expansion of e.g.f. exp(x) * Product_{k>=1} (1 + x^k/k!).

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A327870 Row sums of A327869.

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

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