A007680 -id:A007680 - cp's OEIS frontend

A175925 a(n) = (2n+1)(n+1)!.

1, 6, 30, 168, 1080, 7920, 65520, 604800, 6168960, 68947200, 838252800, 11017036800, 155675520000, 2353813862400, 37922556672000, 648606486528000, 11737685127168000, 224083079700480000, 4500868715126784000

Offset: 0

R. J. Mathar, Oct 19 2010

The denominators of the Taylor expansion coefficients of the double integral d(u) = int_0^1 dx int_0^1 dy exp(-u^2*(x-y)^2) = Sum_{n>=0} (-1)^n*u^(2n)/a(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
D. H. Bailey, J. M. Borwein, R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, eq (18).
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A000142, A005408, A007680.

[(2*n+1)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

Maple
```
A := proc(n) (2*n+1)*(n+1)! ; end proc:
```

Table[(2n+1)(n+1)!,{n,0,20}] (* Harvey P. Dale, Sep 30 2011 *)

a(n) = A005408(n)*A000142(n+1) = (n+1)*A007680(n).
E.g.f.: (1 + 3*x)/(1 - x)^3. - Ilya Gutkovskiy, May 12 2017
From Amiram Eldar, Aug 04 2020: (Start)
Sum_{n>=0} 1/a(n) = sqrt(Pi)*erfi(1) + 1 - e.
Sum_{n>=0} (-1)^n/a(n) = sqrt(Pi)*erf(1) - 1 + 1/e. (End)

A061018 Triangle: a(n,m) = number of permutations of (1,2,...,n) with one or more fixed points in the m first positions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 10, 13, 15, 24, 42, 56, 67, 76, 120, 216, 294, 358, 411, 455, 720, 1320, 1824, 2250, 2612, 2921, 3186, 5040, 9360, 13080, 16296, 19086, 21514, 23633, 25487, 40320, 75600, 106560, 133800, 157824, 179058, 197864, 214551, 229384

Offset: 1

Wouter Meeussen, May 23 2001

Row sums of n are the number of derangements (permutations without fixed point) of n+1, i.e. A000166(n+1).

			For n=3, the permutations are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1); and (x, 2, 3), (x, 3, 2) have a fixed point x in position 1, (x, x, 3), (x, 3, 2), (3, x, 1) have a fixed point x in positions 1 or 2 and (x, x, x), (2, 1, x), (x, 3, 2), (3, x, 1) have a fixed point x in positions 1, 2 or 3, hence {2, 3, 4}
{1},
{1, 1},
{2, 3, 4},
{6, 10, 13, 15},
{24, 42, 56, 67, 76},
{120, 216, 294, 358, 411, 455},
{720, 1320, 1824, 2250, 2612, 2921, 3186}, ...

Columns: A000142, A007680, A002467, A180191.

A061018 := proc(n,m): (n-1)! + add(A061312(n-2,k), k=0..m-2) end: A061312:= proc(n,m): if m=-1 then 0 elif m=0 then n*n! else procname(n,m-1) - procname(n-1,m-1) fi: end: seq(seq(A061018(n,m), m=1..n), n=1..8); # Johannes W. Meijer, Jul 27 2011
T := (n, k) -> `if`(n=k,n!-GAMMA(n+1,-1)/exp(1),n!*(1-hypergeom([-k],[-n],-1))):
for n from 1 to 9 do seq(simplify(T(n,k)), k=1..n) od; # Peter Luschny, Oct 03 2017

Table[Count[Permutations[Range[n]], p_/;( Times@@Take[(p-Range[n]), k]===0)], {n, 7}, {k, n}]

a(n,m) = (n-1)! + Sum_{k=0..m-2} T(n-2, k) where T(n,-1) = 0, T(0,0) = 0, T(n,0) = A001563(n) = n*n!, T(n,m) = T(n,m-1) - T(n-1,m-1) (see A061312).

T(n, k) = n!*(1 - hypergeom([-k], [-n], -1)) for 1 <= k < n and T(n, n) = n! -Gamma(n+1, -1)/exp(1). - Peter Luschny, Oct 03 2017

Edited and information added by Johannes W. Meijer, Jul 27 2011

A069286 Decimal expansion of constant rho satisfying Gaussian Phi(rho * sqrt(2)) = erf(rho) = 1/2.

Original entry on oeis.org

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

Offset: 0

Frank Ellermann, Mar 13 2002

In Bronstein-Semendjajew, Gaussian Phi is the probability integral, i.e., 2 * Normal Distribution Function.

			0.4769362762044698733814183536431305598089697490594706447...

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 6.1.2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Inverse Erf.
Eric Weisstein's World of Mathematics, Normal Distribution Function.
Eric Weisstein's World of Mathematics, Probable Error t= rho * sqrt(2)= 06745...

Cf. A069287 (continued fraction), A007680.

RealDigits[ InverseErf[1/2], 10, 105][[1]] (* Robert G. Wilson v, Oct 11 2004 *)

solve(x=0, 1, erfc(x)-1/2) \\ Charles R Greathouse IV, Oct 15 2015

A082034 a(n) = (4n + 1)n!.

Original entry on oeis.org

1, 5, 18, 78, 408, 2520, 18000, 146160, 1330560, 13426560, 148780800, 1796256000, 23471078400, 330032102400, 4969162598400, 79768136448000, 1359981342720000, 24542432538624000, 467373280518144000, 9366672731480064000

Offset: 0

Paul Barry, Apr 02 2003

A row of the array A082037.

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A082033, A007680.

[(4*n+1)*Factorial(n) : n in [0..20]]; // Vincenzo Librandi, Sep 22 2011

Table[(4n+1)n!,{n,0,20}] (* Harvey P. Dale, Mar 02 2022 *)

a(n) = A016813(n)*n!.
(-4*n+3)*a(n) + n*(4*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 07 2014
4*a(n) + (-4*n-7)*a(n-1) + 3*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 07 2014

A125750 A Moessner triangle using (1, 3, 5, ...).

Original entry on oeis.org

1, 3, 5, 10, 19, 11, 42, 89, 64, 19, 216, 498, 415, 160, 29, 1320, 3254, 3023, 1385, 335, 41, 9360, 24372, 24640, 12803, 3745, 623, 55, 75600, 206100, 223116, 127799, 42938, 8750, 1064, 71, 685440, 1943568, 2227276, 1380076, 516201, 122010, 18354, 1704

Offset: 1

Gary W. Adamson, Dec 06 2006

Right border of the triangle = A028387, left border = A007680.

			Circling the 1, 3, 6, ...(-th) terms in the sequence (1, 3, 5, 7, ...), we get A018387: (1, 5, 11, 19, 29, ...). Taking partial sums of the remaining terms, we get (3, 10, 19, 32, ...) in row 2 and we circle 3 and 19. In row 3 we circle the 10.
First few rows of the triangle are:
    1;
    3,   5;
   10,  19,  11;
   42,  89,  64,  19;
  216, 498, 415, 160,  29;
  ...

J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, pp. 63-64.

Joshua Zucker, Table of n, a(n) for n = 1..78
G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
R. Krebbers, L. Parlant, and A. Silva, Moessner's theorem: an exercise in coinductive reasoning in Coq, Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
Ivan Paasche, Ein neuer Beweis des Moessnerschen Satzes S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
Ivan Paasche, Beweis des Moessnerschen Satzes mittels linearer Transformationen, Arch. Math. (Basel) 6 (1955), 194-199.
Ivan Paasche, Eine Verallgemeinerung des Moessnerschen Satzes, Compositio Math. 12 (1956), 263-270.
Hans Salié, Bemerkung zu einem Satz von A. Moessner, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Cf. A125714, A125751, A125752.

Using "Moessner's Magic" (Conway and Guy, pp. 63-64; cf. A125714), we circle the 1, 3, 6, 10, ...(-th) terms in the sequence (1, 3, 5, 7, ...) and take partial sums of the remaining terms, making row 2. Circle the terms in row 2 one place offset to the left of row 1 terms, then take partial sums. Continue with analogous operations for succeeding rows. The triangle = leftmost circled terms in each row.

More terms from Joshua Zucker, Jun 17 2007

A007683 a(1) = 1; a(n) = -Sum_{k = 1..n-1} (n+k)!a(k)/(2k)!.

Original entry on oeis.org

1, -3, 3, 9, 21, -33, -1173, -13515, -113739, -532209, 6284379, 264830061, 5897799141, 104393462439, 1459983940203, 10308316834293, -308010522508395, -19576840707893409, -726806556195360069, -22261372611370303875, -591210850189999983099

Offset: 1

N. J. A. Sloane

sign

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)

Cf. A067000.

A046900 Triangle inverse to that in A046899.

Original entry on oeis.org

1, -1, 1, 1, -3, 2, 1, 3, -10, 6, -1, 9, 10, -42, 24, -17, 21, 50, 42, -216, 120, -107, -33, 230, 294, 216, -1320, 720, -415, -1173, 670, 1974, 1944, 1320, -9360, 5040, 1231, -13515, -4510, 11130, 17064, 14520, 9360, -75600, 40320, 56671, -113739, -131230, 20202, 136296, 157080, 121680, 75600

Offset: 0

N. J. A. Sloane

Sequence gives numerators; denominators are A001813.

			1; -1/2 1/2; 1/12 -3/12 2/12; ...

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83.

H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)

Cf. A001813, A046899.

with(linalg): b:=proc(n,k) if k<=n then binomial(n+k,k) else 0 fi end: bb:=(n,k)->b(n-1,k-1): B:=matrix(12,12,bb): A:=inverse(B): a:=(n,k)->((2*n-2)!/(n-1)!)*A[n,k]: for n from 0 to 10 do seq(a(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch

max = 10; b[n_, k_] := If[k <= n, Binomial[n+k, k], 0]; BB = Table[b[n, k], {n, 0, max-1}, {k, 0, max-1}]; AA = Inverse[BB]; a[n_, k_] := ((2n-2)!/(n-1)!)*AA[[n, k]]; Flatten[ Table[ a[n, k], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Aug 08 2012, after Emeric Deutsch *)

More terms from Emeric Deutsch, Jun 25 2005

A082035 a(n) = (4n^2+2n+1) * n!.

Original entry on oeis.org

1, 7, 42, 258, 1752, 13320, 113040, 1063440, 11007360, 124467840, 1527724800, 20237817600, 287879961600, 4377595622400, 70875950745600, 1217444836608000, 22115388911616000, 423623726862336000, 8534364149735424000

Offset: 0

Paul Barry, Apr 02 2003

A row of the number array A082038.

Cf. A001564, A082036.

Table[(4n^2+2n+1)n!,{n,0,20}] (* Harvey P. Dale, Jul 15 2011 *)

a(n) = 4*A002775(n) + A007680(n).

A347909 Decimal expansion of Integral_{x=0..1} exp(-x^2) dx.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Sep 18 2021

			0.74682413281242702539946743613185300535449968...

Cf. A019704 (sqrt(Pi)/2 = Integral_{x=0..+oo} exp(-x^2) dx), A002161 (sqrt(Pi) = Integral_{x=-oo..+oo} exp(-x^2) dx).

Cf. A347910 (inverse integrand), A007680.

RealDigits[(Sqrt[Pi]/2) Erf[1], 10, 91][[1]]

intnum(x=0, 1, exp(-x^2)) \\ Michel Marcus, Sep 18 2021

Equals (sqrt(Pi)/2) * erf(1) = (sqrt(Pi)/(2*i)) * erfi(i).

Equals Sum_{k>=0} (-1)^k / ((2*k + 1)*k!). - Ilya Gutkovskiy, Sep 18 2021

A347910 Decimal expansion of Integral_{x=0..1} exp(x^2) dx.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Sep 18 2021

			1.462651745907181608804048586856988155...

Cf. A347909 (inverse integrand), A007680.

RealDigits[(Sqrt[Pi]/2) Erfi[1], 10, 91][[1]]

intnum(x=0, 1, exp(x^2)) \\ Michel Marcus, Sep 18 2021

Equals (sqrt(Pi)/2) * erfi(1) = (sqrt(Pi)/(2*i)) * erf(i).
Equals Sum_{k>=0} 1 / ((2*k + 1)*k!) . - Ilya Gutkovskiy, Sep 18 2021
Equals A019704 * A099288. - R. J. Mathar, Sep 30 2021

cp's OEIS Frontend

A175925 a(n) = (2*n+1)*(n+1)!.

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A061018 Triangle: a(n,m) = number of permutations of (1,2,...,n) with one or more fixed points in the m first positions.

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A069286 Decimal expansion of constant rho satisfying Gaussian Phi(rho * sqrt(2)) = erf(rho) = 1/2.

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A082034 a(n) = (4*n + 1)*n!.

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A125750 A Moessner triangle using (1, 3, 5, ...).

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A007683 a(1) = 1; a(n) = -Sum_{k = 1..n-1} (n+k)!a(k)/(2k)!.

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A046900 Triangle inverse to that in A046899.

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A082035 a(n) = (4n^2+2n+1) * n!.

A175925 a(n) = (2n+1)(n+1)!.

A082034 a(n) = (4n + 1)n!.