A000757 -id:A000757 - cp's OEIS frontend

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

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.

A027765 a(n) = (n+1)*binomial(n+1,5).

Original entry on oeis.org

5, 36, 147, 448, 1134, 2520, 5082, 9504, 16731, 28028, 45045, 69888, 105196, 154224, 220932, 310080, 427329, 579348, 773927, 1020096, 1328250, 1710280, 2179710, 2751840, 3443895, 4275180, 5267241, 6444032, 7832088, 9460704, 11362120, 13571712, 16128189

Offset: 4

Thi Ngoc Dinh (via R. K. Guy)

Number of 7-subsequences of [ 1, n ] with just 1 contiguous pair.

8*a(n) is the number of permutations of (n+1) symbols that 5-commute with an (n+1)-cycle (see A233440 for definition), where 8 = A000757(5). - Luis Manuel Rivera Martínez, Feb 07 2014

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000757, A233440.

[(n+1)*Binomial(n+1,5): n in [4..40]]; // Vincenzo Librandi, Aug 09 2017

a:=n->(sum((numbcomp(n,6)), j=2..n)):seq(a(n), n=6..34); # Zerinvary Lajos, Aug 26 2008

Table[(n+1)Binomial[n+1,5],{n,4,40}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{5,36,147,448,1134,2520,5082},40] (* Harvey P. Dale, Jan 15 2017 *)

G.f.: (5+x)*x^4/(1-x)^7.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=4} 1/a(n) = 5*Pi^2/6 - 575/72.
Sum_{n>=4} (-1)^n/a(n) = 5*Pi^2/12 + 160*log(2)/3 - 2945/72. (End)

Incorrect formula deleted by R. J. Mathar, Feb 13 2016

A027769 a(n) = (n+1)*binomial(n+1, 9).

Original entry on oeis.org

9, 100, 605, 2640, 9295, 28028, 75075, 183040, 413270, 875160, 1755182, 3359200, 6172530, 10943240, 18795370, 31380096, 51074375, 81238300, 126544275, 193393200, 290435145, 429214500, 624962325, 897561600, 1272714300, 1783342704, 2471261100, 3389158080

Offset: 8

Thi Ngoc Dinh (via R. K. Guy)

nonn

Number of 11-subsequences of [ 1, n ] with just 1 contiguous pair.

13208*a(n) is the number of permutations of (n+1) symbols that 9-commute with an (n+1)-cycle (see A233440 for definition), where 13208=A000757(9). - Luis Manuel Rivera Martínez, Feb 06 2014

T. D. Noe, Table of n, a(n) for n = 8..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000757, A233440.

Table[(n+1)*Binomial[n+1, 9], {n, 8, 35}] (* Amiram Eldar, Jan 30 2022 *)

G.f.: (9+x)*x^8/(1-x)^11.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=8} 1/a(n) = 3*Pi^2/2 - 575499/39200.
Sum_{n>=8} (-1)^n/a(n) = 3*Pi^2/4 + 24576*log(2)/35 - 19365109/39200. (End)

A027770 a(n) = (n + 1)*binomial(n + 1, 10).

Original entry on oeis.org

10, 121, 792, 3718, 14014, 45045, 128128, 330616, 787644, 1755182, 3695120, 7407036, 14226212, 26313518, 47070144, 81719000, 138105110, 227779695, 367447080, 580870290, 901350450, 1374917115, 2064391680, 3054514320, 4458356760, 6425278860, 9150726816

Offset: 9

Thi Ngoc Dinh (via R. K. Guy)

Number of 12-subsequences of [ 1, n ] with just one contiguous pair.

120288*a(n) is the number of permutations of (n+1) symbols that 10-commute with an (n+1)-cycle (see A233440 for definition), where 120288 = A000757(10). - Luis Manuel Rivera Martínez, Feb 07 2014

T. D. Noe, Table of n, a(n) for n = 9..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A000757, A233440.

a:= n-> (n+1)*binomial(n+1, 10):
seq(a(n), n=9..36);  # Alois P. Heinz, Oct 04 2019

((# + 1) Binomial[# + 1, 10] &) /@ Range[9, 48] (* Alonso del Arte, Oct 04 2019 *)

G.f.: (10 + x)*x^9/(1 - x)^12.
a(n) = C(n + 1, 10)*C(n + 1, 1). - Zerinvary Lajos, Jun 08 2005, corrected by R. J. Mathar, Feb 13 2016
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=9} 1/a(n) = 5257891/317520 - 5*Pi^2/3.
Sum_{n>=9} (-1)^(n+1)/a(n) = 5*Pi^2/6 + 84992*log(2)/63 - 299498341/317520. (End)

A027771 a(n) = (n+1)*binomial(n+1,11).

Original entry on oeis.org

11, 144, 1014, 5096, 20475, 69888, 210392, 572832, 1436058, 3359200, 7407036, 15519504, 31097794, 59907456, 111435000, 200880160, 352023165, 601277040, 1003321410, 1638819000, 2624841765, 4128783360, 6386711760, 9727323840, 14602906500, 21628990656

Offset: 10

Thi Ngoc Dinh (via R. K. Guy)

Number of 13-subsequences of [ 1, n ] with just 1 contiguous pair.

1214673*a(n) is the number of permutations of (n+1) symbols that 11-commute with an (n+1)-cycle (see A233440 for definition), where 1214673 = A000757(11). - Luis Manuel Rivera Martínez, Feb 07 2014

T. D. Noe, Table of n, a(n) for n = 10..1000
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A000757, A233440.

Table[(n+1)*Binomial[n+1, 11], {n, 10, 35}] (* Amiram Eldar, Jan 30 2022 *)

G.f.: (11+x)*x^10/(1-x)^13.
From Amiram Eldar, Jan 30 2022: (Start)
Sum_{n>=10} 1/a(n) = 11*Pi^2/6 - 57138257/3175200.
Sum_{n>=10} (-1)^n/a(n) = 11*Pi^2/12 + 822272*log(2)/315 - 5773608863/3175200. (End)

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.

A135807 Tenth column (k=9) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 220, 715, 16016, 180180, 2619760, 39503750, 642172960, 11111964864, 204016477080, 3959206825210, 80952590044480, 1739019535313720, 39150661649469744, 921633956154372175, 22640304292494917600

Offset: 0

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

a(n) enumerates circular permutations of {1,2,...,n+9} with exactly nine successor pairs (i,i+1). Due to cyclicity also (n+9,1) is a successor pair.

			a(0)=1 because from the 9!/9 = 40320 circular permutations of n=9 elements only one, namely (1,2,3,4,5,6,7,8,9), has nine successors.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=9.

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. 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, 9], {n, 9, 25}] (* G. C. Greubel, Nov 10 2016 *)

a(n)=((-1)^n + sum( k=0, n-1, (-1)^k * binomial( n, k) * (n - k - 1)!))*binomial(n+9,9) \\ Charles R Greathouse IV, Nov 10 2016

a(n) = binomial(n+9,9)*A000757(n), n>=0.

E.g.f.: (d^9/dx^9) (x^9/9!)*(1-log(1-x))/e^x.

cp's OEIS Frontend

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

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

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A027765 a(n) = (n+1)*binomial(n+1,5).

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A027769 a(n) = (n+1)*binomial(n+1, 9).

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A027770 a(n) = (n + 1)*binomial(n + 1, 10).

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A027771 a(n) = (n+1)*binomial(n+1,11).

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

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A134833 Alternating row sums of triangle A134832.