A055546 -id:A055546 - cp's OEIS frontend

A334383 Decimal expansion of Sum_{k>=0} (-1)^k/(2^k*(k!)^2).

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

Offset: 0

Ilya Gutkovskiy, Apr 25 2020

			1/(2^0*0!^2) - 1/(2^1*1!^2) + 1/(2^2*2!^2) - 1/(2^3*3!^2) + ... = 0.5591341444189799174882684679...

Cf. A055546, A092605.

Bessel function values: A334380 (J(0,1)), this sequence (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselJ[0, Sqrt[2]], 10, 110] [[1]]

besselj(0, sqrt(2)) \\ Michel Marcus, Apr 26 2020

Equals BesselJ(0,sqrt(2)).

Equals BesselI(0,sqrt(2)*i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/(2^0*0!^2) + 1/(2^1*1!^2) + 1/(2^2*2!^2) + 1/(2^3*3!^2) + ... = 1.56608292975635053729238691...

Index entries for sequences related to Bessel functions or polynomials

Cf. A019774, A055546.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), this sequence (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselI[0, Sqrt[2]], 10, 110] [[1]]

suminf(k=0, 1/(2^k*(k!)^2)) \\ Michel Marcus, Apr 26 2020

besseli(0, sqrt(2)) \\ Michel Marcus, Apr 26 2020

Equals BesselI(0,sqrt(2)).

Equals BesselJ(0,sqrt(2)*i). - Jianing Song, Sep 18 2021

A340591 Number A(n,k) of n*(k+1)-step k-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 16, 5, 1, 1, 24, 288, 192, 14, 1, 1, 120, 9216, 24444, 2816, 42, 1, 1, 720, 460800, 7303104, 2738592, 46592, 132, 1, 1, 5040, 33177600, 4234233600, 8204167296, 361998432, 835584, 429, 1, 1, 40320, 3251404800, 4223111040000, 59027412643200, 11332298092032, 53414223552, 15876096, 1430, 1

Offset: 0

Alois P. Heinz, Jan 12 2021

			Square array A(n,k) begins:
  1,  1,     1,         1,              1,                   1, ...
  1,  1,     2,         6,             24,                 120, ...
  1,  2,    16,       288,           9216,              460800, ...
  1,  5,   192,     24444,        7303104,          4234233600, ...
  1, 14,  2816,   2738592,     8204167296,      59027412643200, ...
  1, 42, 46592, 361998432, 11332298092032, 1052109889288796160, ...

Alois P. Heinz, Antidiagonals n = 0..24, flattened

Columns k=0-3 give: A000012, A000108, A006335, A340540.
Rows n=0-2 give: A000012, A000142, |A055546|.
Main diagonal gives A340590.
Cf. A335570.

b:= proc(n, l) option remember; `if`(n=0, 1, (k-> add(
     `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..k)+
     `if`(add(i, i=l)+k x+1, l)), 0))(nops(l)))
    end:
A:= (n, k)-> b(k*n+n, [0$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, l_] := b[n, l] = If[n == 0, 1, Function[k, Sum[
  If[l[[i]]>0, b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, k}]+
  If[Sum[i, {i, l}] + k < n, b[n - 1, Map[#+1&, l]], 0]][Length[l]]];
A[n_, k_] := b[k*n + n, Table[0, {k}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 26 2021, after Alois P. Heinz *)

A088386 a(n) = 2^n*(n!)^3.

Original entry on oeis.org

1, 2, 32, 1728, 221184, 55296000, 23887872000, 16387080192000, 16780370116608000, 24465779630014464000, 48931559260028928000000, 130255810750197006336000000, 450164081952680853897216000000, 1978020976100079672024367104000000, 10855379116837237240069726666752000000

Offset: 0

Cino Hilliard, Nov 08 2003

A010050(n) / a(n) is the probability that there will be no intersections among n rays in the plane with endpoints chosen randomly, uniformly, and independently on a given line segment and angles chosen randomly, uniformly, and independently in [0, 2*Pi). - Jason Zimba, Apr 03 2022

G. C. Greubel, Table of n, a(n) for n = 0..172

Cf. A000165, A010050, A055546.

[2^n*Factorial(n)^3: n in [0..20]]; // G. C. Greubel, Dec 12 2022

Table[2^n*(n!)^3, {n,0,20}] (* G. C. Greubel, Dec 12 2022 *)

PARI
```
for(n=0,20,print1(2^n*(n!)^3, ", "));
```

[2^n*factorial(n)^3 for n in range(21)] # G. C. Greubel, Dec 12 2022

a(0) = 1; a(n) = 2*n^3*a(n-1) for n >= 1. - Georg Fischer, May 23 2021

Offset corrected from 1 to 0 and definition changed by Georg Fischer, May 23 2021

A092170 Sum of squares of alternating factorials : n!^2 - (n-1)!^2 + (n-2)!^2 - ... 1!^2.

Original entry on oeis.org

1, 3, 33, 543, 13857, 504543, 24897057, 1600805343, 130081089057, 13038108350943, 1580312813889057, 227862219988670943, 38547925823643969057, 7561506530728353470943, 1702450746193471070529057

Offset: 1

Christer Mauritz Blomqvist (MauritzTortoise(AT)hotmail.com), Apr 01 2004

The height of a regular simplex (hypertetrahedron) of dimension n and with unit length edges will be h(n)=sqrt(a(n))/n!. The contents (hypervolume) will then be V(n)=V(n-1)*h(n)/n where V(1)=1.

			a(3)=3!^2-a(2)=36-a(2);
a(2)=2!^2-a(1)=4-a(1)=3-1=3 ->
a(3)=36-3=33.

Harvey P. Dale, Table of n, a(n) for n = 1..253

Cf. A005165, A055546.

a[n_] := Sum[(-1)^j*((n - j)!)^2, {j, 0, n - 1}]
Module[{nn=20,fctrls}, fctrls=(Range[nn]!)^2;Table[Total[Times@@@ Partition[ Riffle[Reverse[Take[fctrls,n]],{1,-1},{2,-1,2}],2]], {n, nn}]] (* Harvey P. Dale, Aug 21 2016 *)

a(n) = n!^2 - a(n-1), a(1)=1. - Charles R Greathouse IV, Oct 13 2004

A269943 Triangle read by rows, T(n,k) = ((-1)^k(2n)!/4^k)P[n,k](1/((2n-1)(2n))) where P is the inverse P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 6, 0, 16, 60, 90, 0, 288, 1176, 2520, 2520, 0, 9216, 39360, 98280, 151200, 113400, 0, 460800, 2023296, 5504400, 10311840, 12474000, 7484400, 0, 33177600, 148442112, 426666240, 896575680, 1362160800, 1362160800, 681080400

Offset: 0

Peter Luschny, Mar 27 2016

The P-transform is defined in the link. Compare also the Sage implementation below.

			Triangle starts:
[1]
[0, 1]
[0, 2, 6]
[0, 16, 60, 90]
[0, 288, 1176, 2520, 2520]
[0, 9216, 39360, 98280, 151200, 113400]
[0, 460800, 2023296, 5504400, 10311840, 12474000, 7484400]

Peter Luschny, The P-transform.

Cf. A000680, A055546.

# uses[PtransMatrix from A269941]
eul = lambda n: 1/((2*n-1)*(2*n))
norm = lambda n,k: (-1)^k*factorial(2*n)/4^k
PtransMatrix(7, eul, norm, inverse=True)

T(n,1) = 2^(n-1)*(n-1)!^2 (cf. A055546) for n>=1.

T(n,n) = (2*n)!/2^n = A000680(n).

cp's OEIS Frontend

A334383 Decimal expansion of Sum_{k>=0} (-1)^k/(2^k*(k!)^2).

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A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

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A340591 Number A(n,k) of n*(k+1)-step k-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

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A088386 a(n) = 2^n*(n!)^3.

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Magma

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A092170 Sum of squares of alternating factorials : n!^2 - (n-1)!^2 + (n-2)!^2 - ... 1!^2.

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A269943 Triangle read by rows, T(n,k) = ((-1)^k*(2*n)!/4^k)*P[n,k](1/((2*n-1)*(2*n))) where P is the inverse P-transform, for n>=0 and 0<=k<=n.

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Author

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Programs

Sage

Formula

A269943 Triangle read by rows, T(n,k) = ((-1)^k(2n)!/4^k)P[n,k](1/((2n-1)(2n))) where P is the inverse P-transform, for n>=0 and 0<=k<=n.