A137989 -id:A137989 - cp's OEIS frontend

A137986 Decimal expansion of the number whose Pierce expansion has the sequence of factorial numbers (A000142) as coefficients.

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Feb 26 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric W. Weisstein, Pierce Expansion.
Eric W. Weisstein, Engel Expansion.

Cf. A000142, A137987, A137988, A137989.

P:=proc(n) local a,i,k; a:=0; k:=1; for i from 0 by 1 to n do k:=k*i!; a:=a+(-1)^i/k; print(evalf(a,100)); od; end: P(100);

RealDigits[N[(Sum[(-1)^n*Product[1/((k - 1)!), {k, 1, n}], {n, 1, 250}]), 100]][[1]] (* G. C. Greubel, Jan 01 2017 *)

A137987 Decimal expansion of the inverse of the number whose Engel expansion has the sequence of factorial numbers (A000142) as coefficients.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Feb 26 2008, Apr 18 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Pierce Expansion.
Eric Weisstein's World of Mathematics, Engel Expansion.

Cf. A000142, A137986, A137988, A137989, A287013.

P:=proc(n) local a,i,k; a:=0; k:=1; for i from 0 by 1 to n do k:=k*i!; a:=a+1/k; print(evalf(1/a,100)); od; end: P(100);

RealDigits[N[(1/Sum[Product[1/((k - 1)!), {k, 1, n}], {n, 1, 250}]), 100]][[1]] (* G. C. Greubel, Jan 02 2017 *)

Equals 1/A287013. - Amiram Eldar, Nov 19 2020

A137988 Decimal expansion of the number whose Pierce expansion has the sequence of double factorial numbers (A000165) as coefficients.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Feb 26 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric W. Weisstein, Pierce Expansion.
Eric W. Weisstein, Engel Expansion.

Cf. A000142, A137986, A137987, A137989.

P:=proc(n) local a,i,j,k,w; a:=0; w:=1; for i from 0 by 1 to n do k:=i; j:=i-2; while j>0 do k:=k*j; j:=j-2; od; if (i=0 or i=1) then k:=1; fi; if i=2 then k:=2; fi; w:=w*k; a:=a+(-1)^i/w; print(evalf(a,100)); od; end: P(100);

RealDigits[N[(Sum[(-1)^n*Product[1/((k + 1)!!), {k, 1, n}], {n, 1, 250}]), 100]][[1]] (* G. C. Greubel, Jan 01 2017 *)

A113296 Cumulative product of double factorial A006882.

Original entry on oeis.org

1, 1, 2, 6, 48, 720, 34560, 3628800, 1393459200, 1316818944000, 5056584744960000, 52563198423859200000, 2422112183371431936000000, 327312129899898454671360000000, 211155601241022491077587763200000000

Offset: 0

Jonathan Vos Post, Feb 18 2006

			a(10) = 1!! * 2!! * 3!! * 4!! * 5!! * 6!! * 7!! * 8!! * 9!! * 10!!
= 1 * 2 * 3 * 8 * 15 * 48 * 105 * 384 * 945 * 3840
= 5056584744960000 = 2^23 x 3^9 x 5^4 x 7^2.

Harvey P. Dale, Table of n, a(n) for n = 0..57
Alejandro H. Morales, Igor Pak and Greta Panova, Hook formulas for skew shapes III. Multivariate and product formulas, Algebraic Combinatorics, Vol. 2, No. 5 (2019), pp. 815-861; arXiv preprint, arXiv:1707.00931 [math.CO], 2017-2020.
Eric Weisstein's World of Mathematics, Double Factorial.
Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A006882, A074962, A137988, A137989.

Table[Product[k!!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Table[2^((6n^2+12n+2-3(-1)^n)/24) Pi^(((-1)^n-2n-3)/8) Exp[-1/8] Glaisher^(3/2) BarnesG[(2n+7+(-1)^n)/4] BarnesG[(2n+7-(-1)^n)/4], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 11 2015 *)
FoldList[Times,Range[0,20]!!] (* Harvey P. Dale, Oct 29 2019 *)

a(n) = Product_{k=0..n} k!!.
a(n) = n!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
a(n) = n*(n-2)!! * a(n-1) where a(0) = 0, a(1) = 1 and n >= 2.
a(n) = 2^((6*n^2+12*n+2-3*(-1)^n)/24) * Pi^(((-1)^n-2*n-3)/8) * exp(-1/8) * A^(3/2) * G((2n+7+(-1)^n)/4) * G((2n+7-(-1)^n)/4), where A is the Glaisher-Kinkelin constant (A074962), G(x) is the Barnes G-function. - Vladimir Reshetnikov, Nov 11 2015
Sum_{n>=0} 1/a(n) = 1/A137989. - Amiram Eldar, Nov 09 2020
Sum_{n>=0} (-1)^n/a(n) = A137988. - Amiram Eldar, Apr 12 2021

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A137986 Decimal expansion of the number whose Pierce expansion has the sequence of factorial numbers (A000142) as coefficients.

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A137987 Decimal expansion of the inverse of the number whose Engel expansion has the sequence of factorial numbers (A000142) as coefficients.

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A137988 Decimal expansion of the number whose Pierce expansion has the sequence of double factorial numbers (A000165) as coefficients.

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A113296 Cumulative product of double factorial A006882.

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