A134832 -id:A134832 - cp's OEIS frontend

A135799 Second column (k=1) of triangle A134832 (circular succession numbers).

1, 0, 0, 4, 5, 48, 252, 1832, 14625, 132080, 1323168, 14576076, 175108661, 2278429216, 31920719820, 479088848976, 7669098865441, 130426934203296, 2348478878321248, 44633950190867220, 892899715052136645

Offset: 0

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

a(n) enumerates circular permutations of {1,2,...,n+1} with exactly one successor pair (i,i+1). Due to cyclicity also (n+1,1) is a successor pair.
The o.g.f. of this sequence seems to be the product of the o.g.f. for A000166 (derangements) by the fraction (1+2*x)/(1+x)^2 = 1 - x^2+ 2*x^3 - 3*x^4 + ... = 1 + sum( (-1)^i i x^(i+1), i=0..infinity) - Thomas Baruchel, Jan 08 2016
This correspond to the following transform: a(n) = b(n) - sum((-1)^(n + i) (n - i - 1)*b(i), (i=0..n-2)) - Olivier Gérard, Mar 05 2016

			a(3)=4 because the 4!/4 = 6 circular permutations of n=4 elements (1,2,3,4), (1,4, 3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) have 4,0,1,1, 1 and 1 successor pair(s), respectively.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=1.

G. C. Greubel, Table of n, a(n) for n = 0..448
Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016.

Cf. A000757 (k=0 column), A134515 (k=2 column).

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 1], {n, 21}] (* Michael De Vlieger, Jan 09 2016, after Jean-François Alcover at A134832 *)

a(n) = (n+1)*A000757(n), n>=0.
E.g.f.: (d/dx) x*(1-log(1-x))/e^x.
O.g.f.: see comment section.

A135802 Fifth column (k=4) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 35, 70, 1008, 7560, 75570, 804375, 9443720, 120408288, 1658028645, 24515212540, 387332966720, 6511826843280, 116059273664436, 2185693176650685, 43366955622595920, 904164368153680480

Offset: 0

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

a(n) enumerates circular permutations of {1,2,...,n+4} with exactly four successor pairs (i,i+1). Due to cyclicity also (n+4,1) is a successor pair.

			a(0)=1 because the 4!/4 = 6 circular permutations of n=4 elements (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) have 4,0,1,1,1 and 1 successor pair, respectively. Hence (1,2,3,4) is the only circular permutation with 4 successors.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=4.

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A135801 (column k=3), A135803 (column k=5).

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 4], {n, 4, 25}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = binomial(n+4,4)*A000757(n), n>=0.

E.g.f.: (d^4/dx^4) (x^4/4!)*(1-log(1-x))/e^x.

A135803 Sixth column (k=5) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 56, 126, 2016, 16632, 181368, 2091375, 26442416, 361224864, 5305691664, 83351722636, 1394398680192, 24744942004464, 464237094657744, 9179911341932877, 190814604739422048, 4159156093506930208

Offset: 0

Wolfdieter Lang, Jan 21 2008

a(n) enumerates circular permutations of {1,2,...,n+5} with exactly five successor pairs (i,i+1). Due to cyclicity also (n+5,1) is a successor pair.

			a(0)=1 because from the 5!/5 = 24 circular permutations of n=5 elements only one, namely (1,2,3,4,5), has five successors.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=5.

G. C. Greubel, Table of n, a(n) for n = 0..444

Cf. A135802 (column k=4), A135804 (column k=6).

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 5], {n, 5, 25}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = binomial(n+5,5)*A000757(n), n>=0.

E.g.f.: (d^5/dx^5) (x^5/5!)*(1-log(1-x))/e^x.

A135804 Seventh column (k=6) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 84, 210, 3696, 33264, 392964, 4879875, 66106040, 963266304, 15032793048, 250055167908, 4415595820608, 82483140014880, 1624829831302104, 33659674920420549, 731455984834451184, 16636624374027720832

Offset: 0

Wolfdieter Lang, Jan 21 2008

a(n) enumerates circular permutations of {1,2,...,n+6} with exactly six successor pairs (i,i+1). Due to cyclicity also (n+6,1) is a successor pair.

			a(0)=1 because from the 6!/6 = 120 circular permutations of n=6 elements only one, namely (1,2,3,4,5,6), has six successors.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=6.

G. C. Greubel, Table of n, a(n) for n = 0..444

Cf. A135803 (column k=5), A135805 (column k=7).

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 6], {n, 6, 25}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = binomial(n+6,6)*A000757(n), n>=0.

E.g.f.: (d^6/dx^6) (x^6/6!)*(1-log(1-x))/e^x.

A135805 Eighth column (k=7) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 120, 330, 6336, 61776, 785928, 10456875, 151099520, 2339361024, 38655753552, 678721170036, 12615988058880, 247449420044640, 5106608041235184, 110596074738524661, 2507849090860975488

Offset: 0

Wolfdieter Lang, Jan 21 2008

a(n) enumerates circular permutations of {1,2,...,n+7} with exactly seven successor pairs (i,i+1). Due to cyclicity also (n+7,1) is a successor pair.

			a(0)=1 because from the 7!/7 = 720 circular permutations of n=7 elements only one, namely (1,2,3,4,5,6,7), has seven successors.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=7.

G. C. Greubel, Table of n, a(n) for n = 0..443

Cf. A135804 (column k=6), A135806 (column k=8).

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 7], {n, 7, 25}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = binomial(n+7,7)*A000757(n), n>=0.

E.g.f.: (d^7/dx^7) (x^7/7!)*(1-log(1-x))/e^x.

A135806 Ninth column (k=8) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 165, 495, 10296, 108108, 1473615, 20913750, 321086480, 5263562304, 91807414686, 1696802925090, 33116968654560, 680485905122760, 14681498118551154, 331788224215573983, 7837028408940548400

Offset: 0

Wolfdieter Lang, Jan 21 2008

a(n) enumerates circular permutations of {1,2,...,n+8} with exactly eight successor pairs (i,i+1). Due to cyclicity also (n+8,1) is a successor pair.

			a(0)=1 because from the 8!/8 = 5040 circular permutations of n=8 elements only one, namely (1,2,3,4,5,6,7,8), has eight successors.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=8.

G. C. Greubel, Table of n, a(n) for n = 0..441

Cf. A135805 (column k=7), A135807 (column k=9).

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 8], {n, 8, 25}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = binomial(n+8,8)*A000757(n), n>=0.

E.g.f.: (d^8/dx^8) (x^8/8!)*(1-log(1-x))/e^x.

A135808 Numerator of z-sequence for the Sheffer (Appell type) triangle A134832 (circular succession numbers).

Original entry on oeis.org

0, 0, 1, 1, 8, 8, 159, 659, 6824, 4668, 517581, 4941685, 61043344, 367628164, 10269016939, 147207286503, 2322683458544, 19149119743336, 677630804946393, 2516289402924117, 247342217288517496, 2548438247219028464

Offset: 0

Wolfdieter Lang, Dec 06 2007

The denominators are given in A126352.

For the a- and z-sequences of Sheffer triangles see the W. Lang link under A006232.

			Rationals r(n)=[0,0,1/3, 1/4, 8/5, 8/3, 159/7, 659/8, 6824/9, 4668, 517581/11,...].

Wolfdieter Lang, Rationals r(n).

Cf. A126352 (denominators), A134832.

a(n) = denominator(r(n)) with the rationals r(n) from the given e.g.f.

E.g.f. for rationals r(n): (1-exp(x)/(1-log(1-x)))/x.

A134515 Third column (k=2) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 10, 15, 168, 1008, 8244, 73125, 726440, 7939008, 94744494, 1225760627, 17088219120, 255365758560, 4072255216296, 69021889788969, 1239055874931312, 23484788783212480, 468656477004105810, 9821896865573503095

Offset: 0

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

a(n) enumerates circular permutations of {1,2,...,n+2} with exactly two successor pairs (i,i+1). Due to cyclicity also (n+2,1) is a successor pair.

			a(2)=0 because the 4!/4 = 6 circular permutations of n=4 elements (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) have 4,0,1,1,1 and 1 successor pair, respectively.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=2.

Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016.

Cf. A135799 (column k=1).

E.g.f.: (d^2/dx^2) (x^2/2!)*(1-log(1-x))/e^x.

a(n) = (((n+2)*(n+1))/2)*A000757(n), n>=0.

A134833 Alternating row sums of triangle A134832.

Original entry on oeis.org

1, -1, 1, 0, -2, 12, -16, 144, 368, 4768, 39488, 412288, 4577280, 55671808, 731390976, 10335518720, 156303439872, 2518984822784, 43099089166336, 780268880543744, 14902336357040128, 299452809649520640, 6315501510334480384, 139485953831272710144, 3219718099932104622080

Offset: 0

Wolfdieter Lang, Jan 21 2008

G. C. Greubel, Table of n, a(n) for n = 0..449

Cf. A000142 (factorials as row sums of triangle A134832).

A000757[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := A000757[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[Sum[a[n, k]*(-1)^k, {k, 0, n}], {n, 0, 10}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = Sum_{k=0..n} A134832(n,k)*(-1)^k for n>=0.

A135801 Fourth column (k=3) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 20, 35, 448, 3024, 27480, 268125, 2905760, 34402368, 442140972, 6128803135, 91137168640, 1447072631840, 24433531297776, 437138635330137, 8260372499542080, 164393521482487360, 3436814164696775940

Offset: 0

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

a(n) enumerates circular permutations of {1,2,...,n+3} with exactly three successor pairs (i,i+1). Due to cyclicity also (n+3,1) is a successor pair.

			a(1)=0 because the 4!/4 = 6 circular permutations of n=4 elements (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) have 4,0,1,1,1 and 1 successor pair, respectively.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=3.

G. C. Greubel, Table of n, a(n) for n = 0..446

Cf. A134515 (column k=2), A135802 (column k=4).

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 3], {n, 3, 10}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = binomial(n+3,3)*A000757(n), n>=0.

E.g.f.: (d^3/dx^3) (x^3/3!)*(1-log(1-x))/e^x.

cp's OEIS Frontend

A135799 Second column (k=1) of triangle A134832 (circular succession numbers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A135802 Fifth column (k=4) of triangle A134832 (circular succession numbers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A135803 Sixth column (k=5) of triangle A134832 (circular succession numbers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A135804 Seventh column (k=6) of triangle A134832 (circular succession numbers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A135805 Eighth column (k=7) of triangle A134832 (circular succession numbers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A135806 Ninth column (k=8) of triangle A134832 (circular succession numbers).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A135808 Numerator of z-sequence for the Sheffer (Appell type) triangle A134832 (circular succession numbers).

Views

Author

Keywords

Comments

Examples

Links