A288093 -id:A288093 - cp's OEIS frontend

A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

1, 1, 2, 3, 4, 5, 6, 7, 16, 27, 40, 55, 72, 91, 224, 405, 640, 935, 1296, 1729, 4480, 8505, 14080, 21505, 31104, 43225, 116480, 229635, 394240, 623645, 933120, 1339975, 3727360, 7577955, 13404160, 21827575, 33592320, 49579075, 141639680

Offset: 0

Hugo Pfoertner, Jun 21 2003

nonn

The term "Sextuple factorial numbers" is also used for the sequences A008542, A008543, A011781, A047058, A047657, A049308, which have a different definition. The definition given here is the one commonly used.

			a(14) = 224 because 14*a(14-6) = 14*a(8) = 14*16 = 224.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.

Cf. n!:A000142, n!!:A006882, n!!!:A007661, n!!!!:A007662, n!!!!!:A085157, 6-factorial primes: n!!!!!!+1:A085150, n!!!!!!-1:A051592.

Cf. A288093.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-6);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 21 2019

b:=func< n | n le 6 select n else n*Self(n-6) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 21 2019

a:= n-> `if`(n<1, 1, n*a(n-6)); seq(a(n), n=0..40); # G. C. Greubel, Aug 21 2019

Table[Times@@Range[n,1,-6],{n,0,40}] (* Harvey P. Dale, Aug 10 2019 *)

a(n)=if(n<1, 1, n*a(n-6));
vector(40, n, n--; a(n) ) \\ G. C. Greubel, Aug 21 2019

def a(n):
    if (n<1): return 1
    else: return n*a(n-6)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 21 2019

a(n)=1 for n < 1, otherwise a(n) = n*a(n-6).

Sum_{n>=0} 1/a(n) = A288093. - Amiram Eldar, Nov 10 2020

A143280 Decimal expansion of m(2) = Sum_{n>=0} 1/n!!.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 04 2008

Also decimal expansion of Sum_{n>=1} n!!/n!. - Michel Lagneau, Dec 24 2011

Apart from the first digit, the same as A227569. - Robert G. Wilson v, Apr 09 2014

			3.05940740534257614453947549923327861297725472633534020929971877980544281968...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Penn, Finding the closed form for a double factorial sum, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Double Factorial
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A227569.

Cf. A006882 (n!!), this sequence (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2) )); // G. C. Greubel, Mar 27 2019

RealDigits[ Sqrt[E] + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[ Sum[1/n!!, {n, 0, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)
RealDigits[Total[1/Range[0,200]!!],10,120][[1]] (* Harvey P. Dale, Apr 10 2022 *)

default(realprecision, 100); exp(1/2)*(1 + sqrt(Pi/2)*(1-erfc(1/sqrt(2) ))) \\ G. C. Greubel, Mar 27 2019

numerical_approx(exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=100) # G. C. Greubel, Mar 27 2019

Equals sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)).

A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.298913538088419003401217808261469769077803695683207090885045129...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A007661 (n!!!), A143280 (m(2)), this sequence (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(100)); (Exp(1/3)/3)*(3 + 3^(1/3)*Gamma(1/3, 1/3) + 3^(2/3)*Gamma(2/3, 1/3)); // G. C. Greubel, Mar 27 2019

p = Pochhammer;
m[3] = 5/2 + Sum[1/((-3)^k*((2+3*k)*p[1/3 -k, k])) + 1/((-3)^k*((1+3*k) *p[2/3 -k, k])) +(-1)^(1-k)/(3^k*(k*p[1-k, -1 +k])), {k, 1, Infinity}]
(* or *)
m[3] = (Exp[1/3]/3) (3 + 3^(1/3) (Gamma[1/3] - Gamma[1/3, 1/3]) + 3^(2/3) (Gamma[2/3] - Gamma[2/3, 1/3]));
RealDigits[m[3], 10, 102][[1]]

default(realprecision, 100); (exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - incgam(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - incgam(2/3, 1/3))) \\ G. C. Greubel, Mar 27 2019

numerical_approx((exp(1/3)/3)*(3 + 3^(1/3)*(gamma(1/3) - gamma_inc(1/3, 1/3)) + 3^(2/3)*(gamma(2/3) - gamma_inc(2/3, 1/3))), digits=100) # G. C. Greubel, Mar 27 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288091 Decimal expansion of m(4) = Sum_{n>=0} 1/n!!!!, the 4th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.485944977453557745218809044046404795092682320881969407647249998...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A007662 (n!!!!), A143280 (m(2)), A288055 (m(3)), this sequence (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(100)); (1/4)*Exp(1/4)*(4 + Sqrt(2)* Gamma(1/4, 1/4) + 2*Gamma(1/2, 1/4) + 2*Sqrt(2)*Gamma(3/4, 1/4)) // G. C. Greubel, Mar 28 2019

m[4] = (1/4)*E^(1/4)*(4 + Sqrt[2]*(Gamma[1/4] - Gamma[1/4, 1/4]) + 2*(Sqrt[Pi] - Gamma[1/2, 1/4]) + 2*Sqrt[2]*(Gamma[3/4] - Gamma[3/4, 1/4])); RealDigits[m[4], 10, 104][[1]]

default(realprecision, 100); (1/4)*exp(1/4)*(4+sqrt(2)*(gamma(1/4) - incgam(1/4, 1/4))+2*(sqrt(Pi) -incgam(1/2, 1/4))+2*sqrt(2)*(gamma(3/4) - incgam(3/4, 1/4))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/4)*exp(1/4)*(4 + sqrt(2)*(gamma(1/4) - gamma_inc(1/4, 1/4)) + 2*(sqrt(pi) - gamma_inc(1/2, 1/4)) + 2*sqrt(2)*(gamma(3/4) - gamma_inc(3/4, 1/4))), digits=100) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.640224467733809734176937236963569060632409516968842599452955763...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A085157 (n!5), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), this sequence (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(105)); (1/5)*Exp(1/5)*(5 + (&+[5^(k/5)*Gamma(k/5, 1/5): k in [1..4]])); // G. C. Greubel, Mar 28 2019

m[5] = (1/5)*E^(1/5)*(5 + 5^(1/5)*(Gamma[1/5] - Gamma[1/5, 1/5]) + 5^(2/5)*(Gamma[2/5] - Gamma[2/5, 1/5]) + 5^(3/5)*(Gamma[3/5] - Gamma[3/5, 1/5]) + 5^(4/5)*(Gamma[4/5] - Gamma[4/5, 1/5])); RealDigits[m[5], 10, 102][[1]]

default(realprecision, 105); (1/5)*exp(1/5)*(5 + sum(k=1,4, 5^(k/5)*(gamma(k/5) - incgam(k/5, 1/5)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/5)*exp(1/5)*(5 + sum(5^(k/5)*(gamma(k/5) - gamma_inc(k/5, 1/5)) for k in (1..4))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288094 Decimal expansion of m(7) = Sum_{n>=0} 1/n!7, the 7th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.88695965374084349542856991093670567270530958752016048580439533869...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A114799 (n!7), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), this sequence (m(7)), A288095 (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(105)); (1/7)*Exp(1/7)*(7 + (&+[7^(k/7)*Gamma(k/7, 1/7): k in [1..6]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[7], 10, 104][[1]]
RealDigits[Total[Table[1/Times@@Range[n,1,-7],{n,0,500}]],10,120][[1]] (* Harvey P. Dale, May 21 2023 *)

default(realprecision, 105); (1/7)*exp(1/7)*(7 + sum(k=1,6, 7^(k/7)*(gamma(k/7) - incgam(k/7, 1/7)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/7)*exp(1/7)*(7 + sum(7^(k/7)*(gamma(k/7) - gamma_inc(k/7, 1/7)) for k in (1..6))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288095 Decimal expansion of m(8) = Sum_{n>=0} 1/n!8, the 8th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			3.9892412126901365441336421348019099438359273924576814826209556653...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A114800 (n!8), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), this sequence (m(8)), A288096 (m(9)).

SetDefaultRealField(RealField(107)); (1/8)*Exp(1/8)*(8 + (&+[8^(k/8)*Gamma(k/8, 1/8): k in [1..7]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[8], 10, 103][[1]]

default(realprecision, 105); (1/8)*exp(1/8)*(8 + sum(k=1,7, 8^(k/8)*(gamma(k/8) - incgam(k/8, 1/8)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/8)*exp(1/8)*(8 + sum(8^(k/8)*(gamma(k/8) - gamma_inc(k/8, 1/8)) for k in (1..7))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A288096 Decimal expansion of m(9) = Sum_{n>=0} 1/n!9, the 9th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 05 2017

			4.08137552016889854407110514660961069462641007731860758843485175...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A114806 (n!9), A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), this sequence (m(9)).

SetDefaultRealField(RealField(105)); (1/9)*Exp(1/9)*(9 + (&+[9^(k/9)*Gamma(k/9, 1/9): k in [1..8]])); // G. C. Greubel, Mar 28 2019

m[k_] := (1/k) Exp[1/k] (k + Sum[k^(j/k) (Gamma[j/k] - Gamma[j/k, 1/k]), {j, 1, k - 1}]); RealDigits[m[9], 10, 102][[1]]

default(realprecision, 105); (1/9)*exp(1/9)*(9 + sum(k=1,8, 9^(k/9)*(gamma(k/9) - incgam(k/9, 1/9)))) \\ G. C. Greubel, Mar 28 2019

numerical_approx((1/9)*exp(1/9)*(9 + sum(9^(k/9)*(gamma(k/9) - gamma_inc(k/9, 1/9)) for k in (1..8))), digits=105) # G. C. Greubel, Mar 28 2019

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} k^(j/k)*(gamma(j/k) - gamma(j/k, 1/k))) where gamma(x) is the Euler gamma function and gamma(a,x) the incomplete gamma function.

A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.

Original entry on oeis.org

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

Offset: 1

Bhoris Dhanjal, Feb 26 2021

m(k) can be proved to approach a harmonic series (and diverge) as k approaches infinity.

			4.165243765558384590787262...
For n=10, the series is equal to 1+summation from n=1 to 10 (1/n)=9901/2520.

Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Cf. A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

Multifactorial[n_, k_] := Abs[Apply[Times, Range[-n, -1, k]]]
N[Sum[1/Multifactorial[n, 10], {n, 0, 10000}], 105]
(* or *)
ReciprocalFactorialSumConstant[k_] :=
1/k Exp[1/k] (k + Sum[k^(j/k) Gamma[j/k, 0, 1/k], {j, k - 1}])
N[ReciprocalFactorialSumConstant[10], 105]

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} (k^(j/k)*Gamma(j/k, 1/k))) where Gamma(a,x) the incomplete Gamma function.

cp's OEIS Frontend

A085158 Sextuple factorials, 6-factorials, n!!!!!!, n!6.

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A143280 Decimal expansion of m(2) = Sum_{n>=0} 1/n!!.

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A288055 Decimal expansion of m(3) = Sum_{n>=0} 1/n!!!, the 3rd reciprocal multifactorial constant.

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A288091 Decimal expansion of m(4) = Sum_{n>=0} 1/n!!!!, the 4th reciprocal multifactorial constant.

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A288092 Decimal expansion of m(5) = Sum_{n>=0} 1/n!5, the 5th reciprocal multifactorial constant.

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A288094 Decimal expansion of m(7) = Sum_{n>=0} 1/n!7, the 7th reciprocal multifactorial constant.

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