A347952 -id:A347952 - cp's OEIS frontend

A051683 Triangle read by rows: T(n,k) = n!*k.

1, 2, 4, 6, 12, 18, 24, 48, 72, 96, 120, 240, 360, 480, 600, 720, 1440, 2160, 2880, 3600, 4320, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 80640, 120960, 161280, 201600, 241920, 282240, 322560, 362880, 725760, 1088640, 1451520, 1814400, 2177280, 2540160, 2903040, 3265920

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Numbers with only one nonzero digit when written in factorial base. - Franklin T. Adams-Watters, Nov 28 2011
In other words, numbers m such that A034968(m) = A099563(m). - Antti Karttunen, Jul 02 2013
When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the right within an interval. The subsequence A001563 denotes the circular shifts that start with the first element. Compare A211370 for circular shifts to the left. - Tilman Piesk, Apr 29 2017

			Table begins
   1;
   2,  4;
   6, 12, 18;
  24, 48, 72, 96; ...

Reinhard Zumkeller, Rows n=1..150 of triangle, flattened
Tilman Piesk, Circular shifts to the right (Arrays of permutations).

Cf. A000142, A002024, A002260, row sums give A001286(n+1).

Cf. A001563, A007623, A034968, A099563, A200748, A162608, A347952.

a051683 n k = a051683_tabl !! (n-1) !! (k-1)
a051683_row n = a051683_tabl !! (n-1)
a051683_tabl = map fst $ iterate f ([1], 2) where
   f (row, n) = (row' ++ [head row' + last row'], n + 1) where
     row' = map (* n) row
-- Reinhard Zumkeller, Mar 09 2012

[[Factorial(n)*k: k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jun 15 2015

T[n_, k_] := n!*k; Flatten[Table[T[n, k], {n, 9}, {k, n}]] (* Jean-François Alcover, Apr 22 2011 *)

for(n=1,10, for(k=1,n, print1(n!*k, ", "))) \\ G. C. Greubel, Mar 27 2018

from math import isqrt, factorial, comb
def A051683(n): return factorial(a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(n-comb(a,2)) # Chai Wah Wu, Jun 25 2025

(define (A051683 n) (* (A000142 (A002024 n)) (A002260 n))) ;; Antti Karttunen, Jul 02 2013

T(n,k) = A000142(A002024(n)) * A002260(n,k) = A002024(n)! * A002260(n,k) - Antti Karttunen, Jul 02 2013

Sum_{n>=1} 1/a(n) = e * (gamma - Ei(-1)) = A347952. - Amiram Eldar, Oct 13 2024

A348573 Decimal expansion of exp(-1) * (Ei(1) - gamma).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Oct 23 2021

			0.48482910699568764631040141422173057472446995282397321...

Cf. A001008, A001113, A001620, A002805, A091725, A229837, A283743, A347952.

RealDigits[Exp[-1] (ExpIntegralEi[1] - EulerGamma), 10, 110] [[1]]

(-real(eint1(-1))-Euler)/exp(1) \\ Michel Marcus, Oct 24 2021

Equals Sum_{k>=1} (-1)^(k+1) * H(k) / k!, where H(k) is the k-th harmonic number.

Equals -Integral_{x=0..1} exp(-x)*log(1-x) dx. - Amiram Eldar, Oct 23 2021

A351164 Decimal expansion of gamma * BesselI(0,2) + BesselK(0,2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 03 2022

			1.4297062187372083131867465655452809577372778968399203468724...

Cf. A001008, A001044, A001620, A002805, A070910, A152648, A347952.

RealDigits[EulerGamma BesselI[0, 2] + BesselK[0, 2], 10, 110] [[1]]

Equals Sum_{k>=1} H(k) / (k!)^2, where H(k) is the k-th harmonic number.

A377400 Decimal expansion of e*(gamma - Ei(-1))/2.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Oct 27 2024

			1.08269110766346817971049317424621528419071038...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 120.

Cf. A000142, A001008, A001113, A001620, A005843, A060746, A099285, A347952, A377401.

RealDigits[E(EulerGamma-ExpIntegralEi[-1])/2,10,100][[1]]

Equals Sum_{n>=1} (1/n!)*Sum_{k=1..n} 1/(2*k) (see Shamos).

Equals A347952 / 2.

A353053 Decimal expansion of Pi * BesselY(0,2) / 2 - gamma * BesselJ(0,2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 20 2022

			0.672462966936363624928336197862303184816824730553017...

Cf. A000796, A001008, A001044, A001620, A002805, A091681, A233090, A347952, A348573, A351164.

RealDigits[Pi BesselY[0, 2]/2 - EulerGamma BesselJ[0, 2], 10, 110] [[1]]

Pi*bessely(0,2)/2 - Euler*besselj(0,2) \\ Michel Marcus, Apr 20 2022

Equals Sum_{k>=1} (-1)^(k+1) * H(k) / (k!)^2, where H(k) is the k-th harmonic number.

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A051683 Triangle read by rows: T(n,k) = n!*k.

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A348573 Decimal expansion of exp(-1) * (Ei(1) - gamma).

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A351164 Decimal expansion of gamma * BesselI(0,2) + BesselK(0,2).

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Mathematica

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A377400 Decimal expansion of e*(gamma - Ei(-1))/2.

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A353053 Decimal expansion of Pi * BesselY(0,2) / 2 - gamma * BesselJ(0,2).

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Mathematica

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