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A336633 Triangle read by rows: T(n,k) is the number of generalized permutations related to the degenerate Eulerian numbers with exactly k ascents (0 <= k <= max(0,n-1)).

1, 1, 2, 2, 6, 16, 6, 24, 116, 116, 24, 120, 888, 1624, 888, 120, 720, 7416, 20984, 20984, 7416, 720, 5040, 67968, 270432, 419680, 270432, 67968, 5040, 40320, 682272, 3587904, 7861664, 7861664, 3587904, 682272, 40320, 362880, 7467840, 49701024, 144570624, 204403264, 144570624

Offset: 0

Orli Herscovici, Jul 28 2020

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..max(0,n-1)) begins:
     1;
     1;
     2,     2;
     6,    16,      6;
    24,   116,    116,     24;
   120,   888,   1624,    888,    120;
   720,  7416,  20984,  20984,   7416,   720;
  5040, 67968, 270432, 419680, 270432, 67968, 5040;
  ...

Orli Herscovici, Generalized permutations related to the degenerate Eulerian numbers, arXiv preprint arXiv:2007.13205 [math.CO], 2020.

Columns k = 0..1 give: A000142, A288964. Row sums give A007559.

Cf. A008292, A008517.

Tnk[0, 0] := 1; for n to N do
    for k from 0 to n do if 0 < k and k < n then Tnk[n, k] := (n + k)*Tnk[n - 1, k] + (2*n - k - 1)*Tnk[n - 1, k - 1]; else if k = 0 then Tnk[n, k] := (n + k)*Tnk[n - 1, k]; else Tnk[n, k] := 0; end if; end if; end do;
end do

T(n,k) = (n+k)*T(n-1,k) + (2*n-k-1)*T(n-1,k-1) for positive integers n and 0 <= k < n; T(0,0)=1 (or T(1,0)=1); otherwise T(n,k)=0.
From Peter Bala, Jan 08 2021: (Start)
The following remarks are all conjectures:
The e.g.f. (without the initial 1) A(x,t) = x + (2 + 2*t)*x^2/2! + (6 + 16*t + 6*t^2)*x^3/3! + ... satisfies the autonomous differential equation dA/dx = (1 + A)^2*(1 + t*A)^2.
The series reversion of A(x,t) with respect to x equals Integral_{u = 0..x} 1/((1 + u)^2*(1 + t*u)^2) du.
Let f(x,t) = (1 + x)^2*(1 + t*x)^2 and let D be the operator f(x,t)*d/dx. Then the (n+1)-th row polynomial = D^n(f(x,t)) evaluated at x = 0. (End)

A210493 Transits of Venus since the invention of the telescope by Julian Date (rounded).

Original entry on oeis.org

2317111, 2320030, 2364409, 2367328, 2405867, 2408786, 2453165, 2456085, 2458099, 2497542, 2541921, 2544841, 2583379, 2586298

Offset: 1

Fred Espenak (fred.espenak-1(AT)nasa.gov) or (info01(AT)MrEclipse.com) and Robert G. Wilson v, Jan 23 2013

nonn

"Transits of Venus are among the rarest of predictable astronomical phenomena. They occur in a pattern that generally repeats every 243 years, with pairs of transits eight years apart separated by long gaps of 121.5 years and 105.5 years. The periodicity is a reflection of the fact that the orbital periods of Earth and Venus are close to 8:13 and 243:395 commensurabilities." - Wikipedia

a(n) is approximately 365.25 * A171467(n+46). - Charles R Greathouse IV, Jan 24 2013

			05:19 07 Dec 1631 = 2317110.721528
18:25 04 Dec 1639 = 2320030.267361
05:19 06 Jun 1761 = 2364408.721528
22:25 03 Jun 1769 = 2367328.434028
04:05 09 Dec 1874 = 2405866.670139
17:06 06 Dec 1882 = 2408786.212500
08:19 08 Jun 2004 = 2453164.846528
01:28 06 Jun 2012 = 2456084.561111
02:48 11 Dec 2117 = 2458098.616667
16:01 08 Dec 2125 = 2497542.167361
11:30 11 Jun 2247 = 2541920.979167
04:36 09 Jun 2255 = 2544840.691667
01:40 13 Dec 2360 = 2583378.569444
14:43 10 Dec 2368 = 2586298.113194

Jean Meeus, Transits, Willmann-Bell, 1989.
Jean Meeus, Astronomical Algorithms, Second Ed., 1999.

Fred Espenak, The Ultimate Resource for Eclipse Photography, 2001.
NASA Science, Saturn's Transit of Venus on Dec. 21, 2012
NASA Eclipse Web Site, Planetary Transits Across the Sun
NASA Eclipse Web Site, Eclipses and Transits of 2012
NASA Eclipse Web Site, 2004 and 2012 Transits of Venus
NASA Eclipse Web Site, Six Millennium Catalog of Venus Transits: 2000 BCE to 4000 CE
NASA / Goddard Space Flight Center, Calendars, by L. E. Doggett
The United States Naval Observatory (USNO), Astronomical Applications Department, Julian Date Converter
Wikipedia, 2012 transit of Venus

Cf. A171467.

A130264 Inverse of A014321, which is permutation of positive integers assuming Polignac's conjecture is true.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 13, 8, 9, 10, 12, 14, 15, 16, 17, 11, 18, 24, 21, 20, 19, 31, 23, 26, 22, 28, 32, 30, 29, 27, 33, 36, 34, 37, 25, 44, 39, 38, 48, 40, 46, 35, 49, 45, 51, 53, 41, 50, 43, 57, 56, 52, 62, 58, 42, 47, 68, 54, 60, 63, 71, 59, 65, 67, 55, 72, 70, 64, 73, 75, 76

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), May 20 2007

nonn

Ferenc Adorjan, Table of n,a(n) for n=1,999

Inverse of A014321, Cf. A128754, A128755.

A128754 Permutation of positive integers obtained by swapping n-th natural number with the (n-g)-th sequentially, where g=prime(n+1)-prime(n)-1.

Original entry on oeis.org

4, 3, 6, 9, 8, 11, 1, 2, 12, 15, 16, 13, 18, 7, 5, 21, 30, 23, 22, 17, 19, 25, 10, 27, 34, 29, 32, 31, 20, 24, 36, 37, 42, 39, 46, 47, 33, 26, 35, 41, 44, 43, 28, 53, 48, 51, 50, 49, 62, 55, 56, 61, 66, 38, 14, 59, 52, 45, 68, 63, 40, 67, 64, 57, 72, 71, 70, 73, 74, 77, 80, 79

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), Mar 24 2007

nonn

By numerical explorations up to 50k terms, it seems to be an "infinite braid", i.e. it consists of a single infinite cycle, without any fixed points or closed cycles.

Ferenc Adorjan, Table of n,a(n) for n=1..10000

Inverse of A128755, Cf. A128756, A128757.

{pperm(n)= /* Returns a vector with n terms of the sequence */
local(m,q,v,x,j,ap);j=n+prime(n+6)-prime(n);v=vector(j);x=vector(n);
for(i=1,j,v[i]=i);for(i=1,j,ap=prime(i+1)-prime(i)-1;q=v[i];v[i]=v[i-ap];v[i-ap]=q);
for(i=1,n,x[i]=v[i]);return(x)}

A128755 Inverse of A128754.

Original entry on oeis.org

7, 8, 2, 1, 15, 3, 14, 5, 4, 23, 6, 9, 12, 55, 10, 11, 20, 13, 21, 29, 16, 19, 18, 30, 22, 38, 24, 43, 26, 17, 28, 27, 37, 25, 39, 31, 32, 54, 34, 61, 40, 33, 42, 41, 58, 35, 36, 45, 48, 47, 46, 57, 44, 78, 50, 51, 64, 76, 56, 74, 52, 49, 60, 63, 77, 53, 62, 59, 83, 67, 66, 65

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), Mar 24 2007

nonn

Seemingly the inversion maintains the characteristics of being an "infinite braid".

Ferenc Adorjan, Table of n,a(n) for n=1..10000

Inverse of A128754, Cf. A128756, A128757.

{pinverse(v)= /* Permutation inverse of a positive sequence */
local(n,m,x);n=matsize(v)[2]; x=vector(n);
for(i=1,n,if(v[i]<=n,x[v[i]]=i)); return(x)}
pinverse(A128754)

A129676 Permutation sequence generated by the "odious numbers" (A000069), by swapping n-th natural number by the (n-g)-th sequentially, where g=min(odious(n+1)-odious(n)-1,n-1).

Original entry on oeis.org

3, 1, 5, 4, 6, 2, 9, 7, 10, 12, 11, 8, 15, 13, 17, 16, 18, 20, 19, 14, 23, 21, 24, 22, 27, 25, 29, 28, 30, 26, 33, 31, 34, 36, 35, 32, 39, 37, 40, 38, 43, 41, 45, 44, 46, 48, 47, 42, 51, 49, 53, 52, 54, 50, 57, 55, 58, 60, 59, 56, 63, 61, 65, 64, 66, 68, 67, 62, 71, 69, 72, 70

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), May 01 2007

In contrast to A128754 and A128756 (which are generated analogously from the primes and the lucky numbers, respectively), this sequence seems consisting solely of fixed points and cycles of length 4,5 and 6. It is also notable that the difference of the number of fixed points and the number of cycles never differs by more than 3, up to index 10000, according to numerical tests. Thus the ratio of the number of fixed points to the number of cycles seems to be asymptotically equal to unity.

Inverse of A129677, Cf. A128754, A128756, A129674, A129675, A129680 and A000069.

{vperm(z)=local(n,m,q,v,x,j,g);
/* Permutation of positive integers so that starting with the sequence of positive integers, sequentially swap the i-th term with max(i-g(i),1)-th term, where g(i)=z[i+1]-z[i]-1. */
j=matsize(z)[2]-1;n=j-z[j]+z[j-6];v=vector(j);x=vector(n);for(i=1,j,v[i]=i);
for(i=1,j,g=min(z[i+1]-z[i]-1,i-1);q=v[i];v[i]=v[i-g];v[i-g]=q);for(i=1,n,x[i]=v[i]);return(x)}
a=vperm(A000069)

A129199 Absolute differences of A129198.

Original entry on oeis.org

1, 3, 2, 8, 4, 16, 19, 43, 5, 53, 6, 102, 185, 377, 7, 391, 759, 1527, 9, 1545, 3062, 6134, 10, 6154, 12277, 24565, 11, 24587, 12, 49164, 98291, 196595, 13, 196621, 393202, 786418, 14, 786446, 1572849, 3145713, 15, 3145743, 6291440, 12582896, 12582895

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), Apr 04 2007

nonn

This sequence is known to be a permutation of positive integers, along with its parent sequence (A129198).

In the paper of Slater and Velez it is shown that both this sequence and its parent are permutations of positive integers.

P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, Pacific J. Math., 71, 1977.
William Y. Velez, Research problems 159-160, Discrete Math 110 (1992), pp. 301-302.

This is the absolute difference of A129198, cf. A081145.

A129678 Distance sequence of A129674: A129674(n)-n.

Original entry on oeis.org

2, 0, -2, 2, -1, 1, -2, 2, -1, 2, 0, 1, 2, 0, -6, 2, -1, 2, 0, 1, -4, 2, -1, 1, 2, 0, -4, 2, -1, 1, -2, 2, -1, 2, 0, 1, -4, 2, -1, 1, 2, 0, -4, 2, -1, 2, 0, 1, 2, 0, -6, 2, -1, 1, -2, 2, -1, 2, 0, 1, 2, 0, -6, 2, -1, 2, 0, 1, -4, 2, -1, 1, 2, 0, -4, 2, -1, 2, 0, 1, 2, 0, -6, 2, -1, 1, -2, 2, -1, 2, 0, 1, -4, 2, -1, 1, 2, 0, -4

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), May 01 2007

sign

The numbers in this sequence are in the range of [ -6,2], but they are never -3 nor -5 (tested up to 10^4). The relative density of the values {-6,-4,-2,-1,0,1,2} asymptotically is {1,2,1,4,4,4,8}.

Distance of A129674, Cf. A129679, A129680, A129681 and A001969.

A129679 Distance sequence of A129675: A129675(n)-n.

Original entry on oeis.org

2, 0, -2, 1, 2, -2, -1, 1, 6, -2, 0, -2, -1, 0, -2, 1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0, -2, 1, 2, -2, -1, 1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0, -2, 1, 6, -2, 0, -2, -1, 0, -2, 1, 2, -2, -1, 1, 6, -2, 0, -2, -1, 0, -2, 1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0, -2, 1, 6, -2, 0, -2, -1, 0, -2, 1, 2, -2, -1, 1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), May 01 2007

sign

The numbers in this sequence seem to be in the range of [ -2,6], but they are never 3 nor 5 (tested up to 10^4). The relative density of the values {-2,-1,0,1,2,4,6} asymptotically is {8,4,4,4,1,2,1}.

Distance of A129675, Cf. A129678, A129680, A129681 and A001969.

A129681 Distance sequence of A129677: A129677(n)-n.

Original entry on oeis.org

1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0, -2, 1, 6, -2, 0, -2, -1, 0, -2, 1, 2, -2, -1, 1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0, -2, 1, 2, -2, -1, 1, 6, -2, 0, -2, -1, 0, -2, 1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0, -2, 1, 6, -2, 0, -2, -1, 0, -2, 1, 2, -2, -1, 1, 6, -2, 0, -2, -1, 0, -2, 1, 4, -2, 0, -2, -1, 1, 4, -2, -1, 0, -2, 1, 2, -2, -1, 1, 4, -2

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu or ferencadorjan(AT)gmail.com), May 01 2007

sign

The numbers in this sequence seem to be in the range of [ -2,6], but they are never 3 nor 5 (tested up to 10^4). The relative density of the values {-2,-1,0,1,2,4,6} asymptotically is {8,4,4,4,1,2,1}.

Distance of A129677, Cf. A129678, A129679, A129680 and A000069.

cp's OEIS Frontend

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A336633 Triangle read by rows: T(n,k) is the number of generalized permutations related to the degenerate Eulerian numbers with exactly k ascents (0 <= k <= max(0,n-1)).

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A210493 Transits of Venus since the invention of the telescope by Julian Date (rounded).

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A130264 Inverse of A014321, which is permutation of positive integers assuming Polignac's conjecture is true.

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A128754 Permutation of positive integers obtained by swapping n-th natural number with the (n-g)-th sequentially, where g=prime(n+1)-prime(n)-1.

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PARI

A128755 Inverse of A128754.

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A129676 Permutation sequence generated by the "odious numbers" (A000069), by swapping n-th natural number by the (n-g)-th sequentially, where g=min(odious(n+1)-odious(n)-1,n-1).

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A129199 Absolute differences of A129198.

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A129678 Distance sequence of A129674: A129674(n)-n.

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A129679 Distance sequence of A129675: A129675(n)-n.

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A129681 Distance sequence of A129677: A129677(n)-n.

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