A143820 -id:A143820 - cp's OEIS frontend

A143815 Let A(0)=1, B(0)=0 and C(0)=0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence A(n).

1, 0, 0, 1, 6, 25, 91, 322, 1232, 5672, 32202, 209143, 1432454, 9942517, 69363840, 490303335, 3565609732, 27118060170, 218183781871, 1861370544934, 16729411124821, 156706028787827, 1514442896327792, 14999698898942772, 151838974745743228, 1571513300578303070

Offset: 0

Peter Bala, Sep 03 2008

Compare with A024429 and A024430.
This sequence and its companion sequences B(n) = A143816(n) and C(n) = A143817(n) may be viewed as generalizations of the Bell numbers A000110. Define a sequence R(n) of real numbers by R(n) = Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0, 1, 2, .... It is easy to verify that this sequence satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n, i)*3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2). Some examples are given below.
To find the precise form of the linear relation define two other sequences of real numbers by S(n) = Sum_{k >= 0} (3*k+1)^n/(3*k+1)! and T(n) = Sum_{k >= 0} (3*k+2)^n/(3*k+2)! for n = 0, 1, 2, .... Both S(n) and T(n) satisfy the above recurrence. Then by means of the identities S(n+1) = Sum_{i = 0..n} binomial(n, i)*R(i), T(n+1) = Sum_{i = 0..n} binomial(n, i)*S(i) and R(n+1) = Sum_{i = 0..n} binomial(n, i)*T(i) we obtain the result R(n) = A(n)*R(0) + (B(n) - C(n))*R(1) + C(n)*R(2) = A(n)*R(0) + B(n)*R(1) + C(n)*(R(2) - R(1)) (with corresponding expressions for S(n) and T(n)). This generalizes the Dobinski's relation for the Bell numbers Sum_{k >= 0} k^n/k! = A000110(n)*exp(1).
Some examples of R(n) as a linear combination of R(0), R(1) and R(2) - R(1) are given below. The decimal expansions of R(0) = 1 + 1/3! + 1/6! + 1/9! + ..., R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143819, A143820 and A143821 respectively. Compare with A143628 through A143631.
For n > 0, the number of partitions of {1,2,...,n} into 3,6,9,... classes. - Geoffrey Critzer, Mar 05 2010

			R(n) as a linear combination of R(i), i = 0..2.
  ==================================
  R(n)  |     R(0)    R(1)    R(2)
  ==================================
  R(3)  |       1      -2       3
  R(4)  |       6      -5       7
  R(5)  |      25      -5      16
  R(6)  |      91      20      46
  R(7)  |     322     149     203
  R(8)  |    1232     552    1178
  R(9)  |    5672     991    7242
  R(10) |   32202   -3799   43786
  ...
Column 2 of the above table is A143818.
R(n) as a linear combination of R(0),R(1) and R(2) - R(1).
  =====================================
  R(n)  |     R(0)     R(1)   R(2)-R(1)
  =====================================
  R(3)  |       1        1        3
  R(4)  |       6        2        7
  R(5)  |      25       11       16
  R(6)  |      91       66       46
  R(7)  |     322      352      203
  R(8)  |    1232     1730     1178
  R(9)  |    5672     8233     7242
  R(10) |   32202    39987    43786
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..576
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143816, A143817, A143818, A143819, A143820, A143821.

# (1)
M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[0]:=0:
for n from 1 to M do
b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
c[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
a[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
end do:
A143815:=[seq(a[n], n=0..M)];
# (2)
seq(add(Stirling2(n,3*i),i = 0..floor(n/3)), n = 0..24);
# third Maple program:
b:= proc(n, t) option remember; `if`(n=0, irem(t, 2),
      add(b(n-j, irem(t+1, 3))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 20 2018

a = Exp[x] - 1; f[x_] := 1/3 (E^x + 2 E^(-x/2) Cos[(Sqrt[3] x)/2]); CoefficientList[Series[f[a], {x, 0, 25}], x]*Table[n!, {n, 0, 25}] (* Geoffrey Critzer, Mar 05 2010 *)

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, 1)+Bell_poly(n, w)+Bell_poly(n, w^2))/3; \\ Seiichi Manyama, Oct 13 2022

a(n) = Sum_{k = 0..floor(n/3)} Stirling2(n, 3*k).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(exp(x) - 1).
A143815(n) + A143816(n) + A143817(n) = Bell(n).
E.g.f. is B(A(x)) where A(x) = exp(x) - 1 and B(x) = (1/3)*(exp(x) + 2*exp(-x/2)*cos(sqrt(3)*x/2)). - Geoffrey Critzer, Mar 05 2010
a(n) = ( Bell_n(1) + Bell_n(w) + Bell_n(w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). - Seiichi Manyama, Oct 13 2022
a(n) ~ n^n / (3 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Jun 10 2025

A143817 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence C(n).

Original entry on oeis.org

0, 0, 1, 3, 7, 16, 46, 203, 1178, 7242, 43786, 259634, 1540540, 9414639, 61061613, 428890726, 3266930298, 26581123093, 226393705465, 1986997358251, 17827284972818, 163278469610570, 1531115974317975, 14771302315885372, 147267150734530892, 1521022490460243316

Offset: 0

Peter Bala, Sep 03 2008

Compare with A024429 and A024430.

This sequence and its companion sequences A(n) = A143815 and B(n) = A143816 may be viewed as generalizations of the Bell numbers A000110. Define R(n) = Sum_{k >= 0} (3k)^n/(3k)! for n = 0,1,2,.... Then the real number R(n) is an integral linear combination of R(0) = 1 + 1/3! + 1/6! + ...., R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... . Some examples are given below. The precise result is R(n) = A(n)*R(0) + B(n)*R(1) + C(n)*(R(2)-R(1)). This generalizes the Dobinski relation for the Bell numbers: Sum_{k >= 0} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A143628 through A143631. The decimal expansions of R(0), R(2) - R(1) and R(1) may be found in A143819, A143820 and A143821 respectively.

			R(n) as a linear combination of R(0),R(1)
and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....

Alois P. Heinz, Table of n, a(n) for n = 0..576
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143816, A143818, A143819, A143820, A143821.

# (1)
M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[0]:=0:
for n from 1 to M do
b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
c[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
a[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
end do:
A143817:=[seq(c[n], n=0..M)];
# (2)
seq(add(Stirling2(n,3*i+2),i = 0..floor((n-2)/3)), n = 0..24);
# third Maple program:
b:= proc(n, t) option remember; `if`(n=0, irem(t, 2),
      add(b(n-j, irem(t+1, 3))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 20 2018

a[n_] := Sum[ StirlingS2[n, 3*i+2], {i, 0, (n-2)/3}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Mar 06 2013 *)

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, 1)+w*Bell_poly(n, w)+w^2*Bell_poly(n, w^2))/3; \\ Seiichi Manyama, Oct 13 2022

a(n) = Sum_{k = 0..floor((n-2)/3)} Stirling2(n,3k+2).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(exp(x)-1).
A143815(n) + A143816(n) + A143817(n) = Bell(n).
a(n) = ( Bell_n(1) + w * Bell_n(w) + w^2 * Bell_n(w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). - Seiichi Manyama, Oct 13 2022

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A143816 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence B(n).

Original entry on oeis.org

0, 1, 1, 1, 2, 11, 66, 352, 1730, 8233, 39987, 209793, 1240603, 8287281, 60473869, 463764484, 3647602117, 29165686541, 237499318823, 1984374301872, 17167462137733, 154885317758354, 1461156867801556, 14381004640256202, 146852743814531169, 1546054541191452967

Offset: 0

Peter Bala, Sep 03 2008

Compare with A024429 and A024430.

This sequence and its companion sequences A(n) = A143815 and C(n) = A143817 may be viewed as generalizations of the Bell numbers A000110. Define R(n) = Sum_{k >= 0} (3k)^n/(3k)! for n = 0,1,2,.... Then the real number R(n) is an integral linear combination of R(0) = 1 + 1/3! + 1/6! + ...., R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... and R(1) = 1/2! + 1/5! + 1/8! + .... Some examples are given below. The precise result is R(n) = A(n)*R(0) + B(n)*R(1) + C(n)*(R(2)-R(1)). This generalizes the Dobinski relation for the Bell numbers: Sum_{k >= 0} k^n/k! = A000110(n)*exp(1). See A143815 for more details. Compare with A143628 through A143631. The decimal expansions of R(0), R(2) - R(1) and R(1) may be found in A143819, A143820 and A143821 respectively.

			R(n) as a linear combination of R(0),R(1)
and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....

Alois P. Heinz, Table of n, a(n) for n = 0..576
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A000110, A024429, A024430, A143628, A143629, A143630, A143631, A143815, A143817, A143818, A143819, A143820, A143821.

# (1)
M:=24: a:=array(0..100): b:=array(0..100): c:=array(0..100):
a[0]:=1: b[0]:=0: c[0]:=0:
for n from 1 to M do
b[n]:=add(binomial(n-1,k)*a[k], k=0..n-1);
c[n]:=add(binomial(n-1,k)*b[k], k=0..n-1);
a[n]:=add(binomial(n-1,k)*c[k], k=0..n-1);
end do:
A143816:=[seq(b[n], n=0..M)];
# (2)
seq(add(Stirling2(n,3*i+1),i = 0..floor((n-1)/3)), n = 0..24);
# third Maple program:
b:= proc(n, t) option remember; `if`(n=0, irem(t, 2),
      add(b(n-j, irem(t+1, 3))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 20 2018

m = 23; a[0] = 1; b[0] = 0; c[0] = 0; For[n = 1, n <= m, n++, b[n] = Sum[Binomial[n - 1, k]*a[k], {k, 0, n - 1}]; c[n] = Sum[Binomial[n - 1, k]*b[k], {k, 0, n - 1}]; a[n] = Sum[Binomial[n - 1, k]*c[k], {k, 0, n - 1}]]; A143816 = Table[ b[n], {n, 0, m}] (* Jean-François Alcover, Mar 06 2013, after Maple *)

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, 1)+w^2*Bell_poly(n, w)+w*Bell_poly(n, w^2))/3; \\ Seiichi Manyama, Oct 13 2022

a(n) = Sum_{k = 0..floor((n-1)/3)} Stirling2(n,3k+1).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(exp(x)-1). A143815(n) + A143816(n) + A143817(n) = Bell(n).
a(n) = ( Bell_n(1) + w^2 * Bell_n(w) + w * Bell_n(w^2) )/3, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/3). - Seiichi Manyama, Oct 13 2022

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

Original entry on oeis.org

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

Offset: 1

Peter Bala, Sep 03 2008

Previous name was: Decimal expansion of the constant 1 + 1/3! + 1/6! + 1/9! + ... = 1.16805 83133 75918 ... .
Define a sequence R(n) of real numbers by R(n) := Sum_{k>=0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(0); the decimal expansions of R(2) - R(1) = 1/1! + 1/4! + 1/7! and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143820 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i=0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.
Bowman and Mc Laughlin (Corollary 10 with m = -1) give a continued fraction expansion for this constant and deduce the constant is irrational. - Peter Bala, Apr 17 2017

			1.168058313375918525516256929611144747716933295113292516385891232685...
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
=======================================
  R(n)  |     R(0)     R(1)   R(2)-R(1)
=======================================
  R(3)  |       1        1        3
  R(4)  |       6        2        7
  R(5)  |      25       11       16
  R(6)  |      91       66       46
  R(7)  |     322      352      203
  R(8)  |    1232     1730     1178
  R(9)  |    5672     8233     7242
  R(10) |   32202    39987    43786
  ...
The column entries are from A143815, A143816 and A143817.

D. Bowman and J. Mc Laughlin, Polynomial continued fractions, arXiv:1812.08251 [math.NT], 2018; Acta Arith. 103 (2002), no. 4, 329-342.
Michael Penn, Two sum identities, YouTube video, 2020.
Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143820, A143821.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), this sequence (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

RealDigits[ N[ 1/3*(2*Cos[Sqrt[3]/2]/Sqrt[E] + E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
With[{nn=120},RealDigits[N[Total[Table[1/(3n)!,{n,nn}]]+1,nn],10,nn][[1]]] (* Harvey P. Dale, Apr 20 2013 *)

suminf(k=0, 1/(3*k)!) \\ Michel Marcus, Feb 21 2016

Equals (exp(1) + exp(w) + exp(w^2))/3, where w = exp(2*Pi*i/3).
A143819 + A143820 + A143821 = exp(1).
Equals 1/3 * (e + 2 * cos(sqrt(3)/2) / sqrt(e)). - Bernard Schott, Mar 01 2020
Sum_{k>=0} (-1)^k / (3*k)! = (exp(-1) + 2*exp(1/2)*cos(sqrt(3)/2))/ 3 = 0.83471946857721... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(6 - 6/(121 - 120/(505 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n )*(3*n - 1)*(3*n - 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

Offset corrected by R. J. Mathar, Feb 05 2009

New name from Bernard Schott, Mar 02 2020

A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

Original entry on oeis.org

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

Offset: 0

Peter Bala, Sep 03 2008

Define a sequence of real numbers R(n) by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2... . This constant is R(1); the decimal expansions of R(0) = 1 + 1/3!+ 1/6! + 1/9! + ... and R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... may be found in A143819 and A143820. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n,i) *3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.

			R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....
...
The column entries are from A143815, A143816 and A143817.

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 460.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143820, A346441.

RealDigits[ N[ -((Cos[Sqrt[3]/2] - E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E])), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Constant = (exp(1) + w*exp(w) + w^2*exp(w^2))/3, where w = exp(2*Pi*i/3). A143819 + A143820 + A143821 = exp(1).

Continued fraction: 1/(2 - 2/(61 - 60/(337 - 336/(991 - ... - P(n-1)/((P(n) + 1) - ... ))))), where P(n) = (3*n)*(3*n + 1)*(3*n + 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

Offset corrected by R. J. Mathar, Feb 05 2009

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.

Original entry on oeis.org

1, 25, 5251, 3780721, 6487717237, 21798729916321, 126737815733490295, 1171057417377450325801, 16160592359808814496053801, 317652603424402057734433512457, 8567090714356123497097671830965291, 307592825008242258039794809418977808065

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A100089, A143820, A330044, A337726, A337727, A337728, A337729, A337730.

Table[(3 n + 1)! Sum[1/(3 k + 1)!, {k, 0, n}], {n, 0, 11}]
Table[(3 n + 1)! SeriesCoefficient[(Exp[3 x/2] - 2 Sin[Pi/6 - Sqrt[3] x/2])/(3 Exp[x/2] (1 - x^3)), {x, 0, 3 n + 1}], {n, 0, 11}]
Table[Floor[(Exp[3/2] + 2 Sin[(3 Sqrt[3] - Pi)/6])/(3 Sqrt[Exp[1]]) (3 n + 1)!], {n, 0, 11}]

a(n) = (3*n+1)!*sum(k=0, n, 1/(3*k+1)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (exp(3*x/2) - 2 * sin(Pi/6 - sqrt(3)*x/2)) / (3*exp(x/2) * (1 - x^3)) = x + 25*x^4/4! + 5251*x^7/7! + 3780721*x^10/10! + ...

a(n) = floor(c * (3*n+1)!), where c = (exp(3/2) + 2 * sin((3 * sqrt(3) - Pi) / 6))/(3 * sqrt(exp(1))) = A143820.

A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

			1/1! + 1/5! + 1/9! + ... = 1.008336089225848981767442...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 36.

Cf. A001113, A049469, A073742, A143820, A332890, A334364, A334365.

RealDigits[(Sin[1] + Sinh[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+1)!) \\ Michel Marcus, Apr 25 2020

Equals (sin(1) + sinh(1))/2.

A100089 a(n) = (3*n+1)!.

Original entry on oeis.org

1, 24, 5040, 3628800, 6227020800, 20922789888000, 121645100408832000, 1124000727777607680000, 15511210043330985984000000, 304888344611713860501504000000, 8222838654177922817725562880000000

Offset: 0

Jorge Coveiro, Dec 26 2004

nonn

Cf. A000142, A143820.

Maple
```
seq( (3*n+1)!,n=0..13);
```

Table[(3*n+1)!, {n, 0, 10}] (* Amiram Eldar, Jun 23 2020 *)

Sum_{n>=0} 1/a(n) = A143820. - Amiram Eldar, Jun 23 2020

Offset 0 and a(0) added by Amiram Eldar, Jun 23 2020

A349088 a(n) = n! * Sum_{k=0..floor((n-1)/3)} 1 / (3*k+1)!.

Original entry on oeis.org

0, 1, 2, 6, 25, 125, 750, 5251, 42008, 378072, 3780721, 41587931, 499055172, 6487717237, 90828041318, 1362420619770, 21798729916321, 370578408577457, 6670411354394226, 126737815733490295, 2534756314669805900, 53229882608065923900, 1171057417377450325801

Offset: 0

Ilya Gutkovskiy, Mar 25 2022

nonn

Cf. A002627, A009628, A143820, A186763, A337725, A349089, A352659.

Table[n! Sum[1/(3 k + 1)!, {k, 0, Floor[(n - 1)/3]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Exp[x] - 2 Exp[-x/2] Sin[(Pi - 3 Sqrt[3] x)/6])/(3 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: (exp(x) - 2 * exp(-x/2) * sin((Pi - 3*sqrt(3)*x)/6)) / (3*(1 - x)).

a(n) = floor(c * n!) for n > 0, where c = 1.041865355... = A143820.

A365066 Decimal expansion of the constant 1/0! - 1/1! + 1/2! + 1/3! - 1/4! + 1/5! + 1/6! - 1/7! + ...

Original entry on oeis.org

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

Offset: 0

Peter McNair, Aug 19 2023

			0.63455111826122554275761424130960772236307995025163265587548911687697314...

Cf. A143820.

Digits:=105: evalf(sum(1/(3*n)!-1/(3*n+1)!+1/(3*n+2)!, n=0..infinity)); # Michal Paulovic, Aug 20 2023

RealDigits[E/3 - (4*Sin[Sqrt[3]/2-Pi/6])/(3*Sqrt[E]), 10, 105][[1]]

suminf(n=0,1/(3*n)!-1/(3*n+1)!+1/(3*n+2)!) \\ Michal Paulovic, Aug 20 2023

Equals e - 2*A143820.
Equals Sum_{n>=0} (-1)^(2^((n-1) mod 3) mod 2) / n! = e/3 - 4*sin(sqrt(3)/2 - Pi/6) / (3*sqrt(e)).
Equals Sum_{n>=0} 1/(3*n)! - 1/(3*n+1)! + 1/(3*n+2)!. - Michal Paulovic, Aug 19 2023

cp's OEIS Frontend

A143815 Let A(0)=1, B(0)=0 and C(0)=0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence A(n).

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A143817 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence C(n).

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A143816 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)*B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence B(n).

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A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

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A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

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A337725 a(n) = (3*n+1)! * Sum_{k=0..n} 1 / (3*k+1)!.

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A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

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A143815 Let A(0)=1, B(0)=0 and C(0)=0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)*C(k). This entry gives the sequence A(n).

A143817 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence C(n).

A143816 Let A(0) = 1, B(0) = 0 and C(0) = 0. Let B(n+1) = Sum_{k = 0..n} binomial(n,k)* A(k), C(n+1) = Sum_{k = 0..n} binomial(n,k)B(k) and A(n+1) = Sum_{k = 0..n} binomial(n,k)C(k). This entry gives the sequence B(n).

A337725 a(n) = (3n+1)! Sum_{k=0..n} 1 / (3*k+1)!.