A000453 -id:A000453 - cp's OEIS frontend

A056328 Number of reversible string structures with n beads using exactly four different colors.

0, 0, 0, 1, 6, 37, 183, 877, 3930, 17185, 73095, 306361, 1267266, 5198557, 21182343, 85910917, 347187210, 1399451545, 5629911015, 22616256721, 90754855026, 363890126677, 1458172596903, 5840531635357, 23385650196090

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly four different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

			For a(5)=6, the color patterns are ABCDA, ABCBD, AABCD, ABACD, ABCAD, and ABBCD. The first two are achiral. - _Robert A. Russell_, Oct 14 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (8, -12, -44, 121, 12, -228, 144).

Column 4 of A284949.
Cf. A056311.
Cf. A000453 (oriented), A320527 (chiral), A304974 (achiral).

k=4; Table[(StirlingS2[n,k] + If[EvenQ[n], StirlingS2[n/2+2,4] - StirlingS2[n/2+1,4] - 2StirlingS2[n/2,4], 2StirlingS2[(n+3)/2,4] - 4StirlingS2[(n+1)/2,4]])/2, {n,30}] (* Robert A. Russell, Oct 14 2018 *)
Ach[n_, k_] := Ach[n, k] = If[n < 2, Boole[n == k && n >= 0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
k = 4; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 1, 6, 37, 183}, 30] (* Robert A. Russell, Oct 14 2018 *)

a(n) = A056323(n) - A001998(n-1).
Empirical g.f.: -x^4*(3*x^3 + x^2 - 2*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(3*x^2-1)). - Colin Barker, Nov 25 2012
From Robert A. Russell, Oct 14 2018: (Start)
a(n) = (S2(n,k) + A(n,k))/2, where k=4 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
a(n) = (A000453(n) + A304974(n)) / 2 = A000453(n) - A320527(n) = A320527(n) + A304974(n). (End)

A062255 4th level triangle related to Eulerian numbers and binomial transforms (A062254 is third level, A062253 is second level, triangle of Eulerian numbers is first level and triangle with Z(0,0)=1 and Z(n,k)=0 otherwise is 0th level).

Original entry on oeis.org

1, 10, 0, 65, 20, 0, 350, 350, 35, 0, 1701, 3696, 1316, 56, 0, 7770, 30660, 24570, 4200, 84, 0, 34105, 220620, 325620, 131020, 12195, 120, 0, 145750, 1447050, 3513345, 2656720, 613140, 33330, 165, 0, 611501, 8901992, 33074448, 41503484, 18444833, 2634192, 87406, 220, 0

Offset: 0

Henry Bottomley, Jun 14 2001

Binomial transform of n^4*k^n is ((kn)^4 + 6(kn)^3 + (7 - 4k)(kn)^2 + (1 - 4k + k^2)(kn))*(k + 1)^(n - 4); of n^5*k^n is ((kn)^5 + 10(kn)^4 + (25 - 10k)(kn)^3 + (15 - 30k + 5k^2)(kn)^2 + (1 - 11k + 11k^2 - k^3)(kn))*(k + 1)^(n - 5); of n^6*k^n is ((kn)^6 + 15(kn)^5 + (65 - 20k)(kn)^4 + (90 - 120k + 15k^2)(kn)^3 + (31 - 146k + 91k^2 - 6k^3)(kn)^2 + (1 - 26k + 66k^2 - 26k^3 + k^4)(kn))*(k + 1)^(n - 6). This sequence gives the (unsigned) polynomial coefficients of (kn)^4.

			Rows start:
 (1),
 (10,0),
 (65,20,0),
 (350,350,35,0), etc.

First column is A000453. Diagonals include A000007 and all but the start of A000292. Row sums are A000454. Taking all the levels together to create a pyramid, one face would be A010054 as a triangle with a parallel face which is Pascal's triangle (A007318) with two columns removed, another face would be a triangle of Stirling numbers of the second kind (A008277) and a third face would be A000007 as a triangle, with a triangle of Eulerian numbers (A008292), A062253, A062254 and A062255 as faces parallel to it. The row sums of this last group would provide a triangle of unsigned Stirling numbers of the first kind (A008275).

E(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, (k+1)*E(n-1, k)+(n-k)*E(n-1, k-1)));
A2(n, k) = if ((n<0) || (k<0), 0, (k+2)*A2(n-1, k)+(n-k)*A2(n-1, k-1)+E(n, k));
A3(n, k) = if ((n<0) || (k<0), 0, (k+3)*A3(n-1, k)+(n-k)*A3(n-1, k-1) + A2(n, k));
A4(n, k) = if ((n<0) || (k<0), 0, (k+4)*A4(n-1, k)+(n-k)*A4(n-1, k-1)+ A3(n, k));
row4(n) = vector(n+1, k, A4(n,k-1)); \\ Michel Marcus, Jan 27 2025

A(n, k) = (k+4)*A(n-1, k)+(n-k)*A(n-1, k-1) + A062254(n, k).

More terms from Michel Marcus, Jan 27 2025

A016094 Expansion of 1/((1-9x)(1-10x)(1-11x)(1-12*x)).

Original entry on oeis.org

1, 42, 1105, 23310, 431221, 7309722, 116419465, 1769717670, 25948716541, 369730963602, 5147200519825, 70298695224030, 944897655707461, 12530341519244682, 164265473257148185, 2132247784185258390

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (42,-659,4578,-11880)

Cf. A000453, A025211, A028025, A003468, A028165, A028200, A016109, A016075.

CoefficientList[Series[1/((1-9x)(1-10x)(1-11x)(1-12x)) ,{x,0,20}],x] (* or *) LinearRecurrence[{42,-659,4578,-11880},{1,42,1105,23310},20] (* Harvey P. Dale, Dec 14 2021 *)

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,9), n >= 3. - Milan Janjic, Apr 26 2009
a(n) = 42*a(n-1) - 659*a(n-2) + 4578*a(n-3) - 11880*a(n-4), n >= 4. - Vincenzo Librandi, Mar 18 2011
a(n) = 23*a(n-1) - 132*a(n-2) + 10^(n+1) - 9^(n+1), n >= 2. - Vincenzo Librandi, Mar 18 2011
a(n) = 5*10^(n+2) + 2*12^(n+2) - 11^(n+3)/2 - 3*9^(n+2)/2. - R. J. Mathar, Mar 19 2011

A016109 Expansion of 1/((1-7x)(1-8x)(1-9x)(1-10*x)).

Original entry on oeis.org

1, 34, 725, 12410, 186501, 2571114, 33339685, 413066170, 4941549701, 57504755594, 654463491045, 7314256515930, 80522026412101, 875355238834474, 9415203971344805, 100355146006589690

Offset: 0

Robert G. Wilson v

nonn

Index entries for linear recurrences with constant coefficients, signature (34,-431,2414,-5040).

Cf. A000453, A025211, A028025, A003468, A028165, A028200, A016075, A016094.

CoefficientList[Series[1/((1-7x)(1-8x)(1-9x)(1-10x)),{x,0,20}],x] (* or *) LinearRecurrence[{34,-431,2414,-5040},{1,34,725,12410},21] (* Harvey P. Dale, Jan 26 2012 *)

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,7), n >= 3. - Milan Janjic, Apr 26 2009; adapted by R. J. Mathar, Mar 15 2011
a(n) = 19*a(n-1) - 90*a(n-2) + 8^(n+1) - 7^(n+1), n >= 2. - Vincenzo Librandi, Mar 12 2011
a(n) = (10^(n+3) - 3*9^(n+3) + 3*8^(n+3) - 7^(n+3))/6.  - Bruno Berselli, Mar 12 2011
a(n) = 34*a(n-1) - 431*a(n-2) + 2414*a(n-3) - 5040*a(n-4); a(0)=1, a(1)=34, a(2)=725, a(3)=12410. - Harvey P. Dale, Jan 26 2012

Offset changed to 0 by Vincenzo Librandi, Mar 12 2011

A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 36, 750, 11880, 159027, 1899612, 20912320, 216627840, 2141764053, 20415995028, 189036065010, 1709751003480, 15170932662679, 132511015347084, 1142399079991620, 9741955019900400, 82318282158320505, 690223721118368580, 5749622251945664950

Offset: 8

Wolfdieter Lang

See A000771.

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049435. a(n)= A008277(n, 8).

lst={};Do[f=StirlingS2[n, 8];AppendTo[lst, f], {n, 8, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)

G.f.: x^8/product_{k=1..8} (1-k*x).
E.g.f.: ((exp(x)-1)^8)/8!.
a(n) = det(|s(i+8,j+7)|, 1 <= i,j <= n-8), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A049435 Stirling numbers of second kind: 10th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 55, 1705, 39325, 752752, 12662650, 193754990, 2758334150, 37112163803, 477297033785, 5917584964655, 71187132291275, 835143799377954, 9593401297313460, 108254081784931500, 1203163392175387500, 13199555372846848005, 143197070509423605675

Offset: 10

Wolfdieter Lang

See A000771.

Index entries for linear recurrences with constant coefficients, signature (55,-1320,18150,-157773,902055,-3416930,8409500,-12753576,10628640,-3628800).

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434, A049447. a(n) = A008277(n, 10).

lst={};Do[f=StirlingS2[n, 10];AppendTo[lst, f], {n, 10, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x) (1 - 9 x) (1 - 10 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
StirlingS2[Range[10,35],10] (* Harvey P. Dale, Nov 07 2020 *)

G.f.: x^10/Product_{k=1..10} (1-k*x).
E.g.f.: ((exp(x)-1)^10)/10!.
a(n) = det(|s(i+10,j+9)|, 1 <= i,j <= n-10), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A049447 Stirling numbers of second kind: 9th column of Stirling2 triangle A008277.

Original entry on oeis.org

1, 45, 1155, 22275, 359502, 5135130, 67128490, 820784250, 9528822303, 106175395755, 1144614626805, 12011282644725, 123272476465204, 1241963303533920, 12320068811796900, 120622574326072500, 1167921451092973005, 11201516780955125625, 106563273280541795575

Offset: 9

Wolfdieter Lang

See A000771.

Cf. A000225, A000392, A000453, A000481, A000770, A000771, A049434.

lst={};Do[f=StirlingS2[n, 9];AppendTo[lst, f], {n, 9, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x) (1 - 6 x) (1 - 7 x) (1 - 8 x) (1 - 9 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
StirlingS2[Range[9,30],9] (* Harvey P. Dale, Dec 12 2022 *)

a(n)= A008277(n, 9).
G.f.: x^9/product_{k=1..9} (1-k*x).
E.g.f.: ((exp(x)-1)^9)/9!.
a(n) = det(|s(i+9,j+8)|, 1 <= i,j <= n-9), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 4, 28, 167, 824, 3840, 16920, 72655, 305140, 1265264, 5193188, 21173607, 85887984, 347150080, 1399355440, 5629755935, 22615859180, 90754215024, 363888497148, 1458169977847, 5840524999144, 23385639542720, 93613165023560, 374664497695215, 1499293455643620, 5999080285068784, 24002040333605908

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(5)=4, the chiral pairs are AABCD-ABCDD, ABACD-ABCDC, ABBCD-ABCCD and ABCAD-ABCDB.

Index entries for linear recurrences with constant coefficients, signature (8, -12, -44, 121, 12, -228, 144).

Col. 4 of A320525.

Cf. A000453 (oriented), A056328 (unoriented), A304974 (achiral).

k=4; Table[(StirlingS2[n,k] - If[EvenQ[n], StirlingS2[n/2+2,4] - StirlingS2[n/2+1,4] - 2StirlingS2[n/2,4], 2StirlingS2[(n+3)/2,4] - 4StirlingS2[(n+1)/2,4]])/2, {n,30}]
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k = 4; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 0, 4, 28, 167}, 30]

a(n) = (S2(n,k) - A(n,k))/2, where k=4 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].
G.f.: (x^4 / Product_{k=1..4} (1 - k*x) - x^4*(1 + x)^2*(1 - 2 x^2) / Product_{k=1..4} (1 - k*x^2)) / 2.
a(n) = (A000453(n) - A304974(n)) / 2 = A000453(n) - A056328(n) = A056328(n) - A304974(n).

A346976 Expansion of e.g.f. log( 1 + (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1666, 6510, 7855, -270050, -4942894, -63052990, -682650605, -6309889950, -42960995804, 348211510, 7739540496935, 202902567668150, 3863986259609686, 61527382177040010, 807717870749781475, 7066953051021894250, -33781117662453993424

Offset: 4

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000453, A327505, A346954, A346974, A346975, A346977.

Cf. A346946, A346954.

nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

a(n) = Stirling2(n,4) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ -(n-1)! * 2^(n+1) * cos(n*arctan((2*arctan(1/(1 + 1/6^(1/4)))) / log(1 + 2*6^(1/4) + 2*6^(1/2)))) / (4*arctan(1/(1 + 1/6^(1/4)))^2 + log(1 + 2*6^(1/4) + 2*6^(1/2))^2)^(n/2). - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (-1)^(k-1) * (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A056280 Number of primitive (aperiodic) word structures of length n which contain exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 1, 10, 65, 350, 1700, 7770, 34095, 145750, 611435, 2532530, 10391395, 42355940, 171797200, 694337290, 2798799150, 11259666950, 45232081795, 181509069700, 727778478075, 2916342574750, 11681056021300, 46771289738800, 187226354413735, 749329038527580

Offset: 1

Marks R. Nester

nonn

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column 4 of A137651.

Cf. A056269.

a(n) = Sum_{n > 0, d|n} mu(d)*A000453(n/d).

G.f.: Sum_{k>=1} mu(k) * x^(4*k) / Product_{j=1..4} (1 - j*x^k). - Ilya Gutkovskiy, Apr 15 2021

Terms a(25) and beyond from Andrew Howroyd, Apr 15 2021

cp's OEIS Frontend

A056328 Number of reversible string structures with n beads using exactly four different colors.

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A062255 4th level triangle related to Eulerian numbers and binomial transforms (A062254 is third level, A062253 is second level, triangle of Eulerian numbers is first level and triangle with Z(0,0)=1 and Z(n,k)=0 otherwise is 0th level).

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A016094 Expansion of 1/((1-9*x)*(1-10*x)*(1-11*x)*(1-12*x)).

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A016109 Expansion of 1/((1-7*x)*(1-8*x)*(1-9*x)*(1-10*x)).

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A049434 Stirling numbers of second kind: 8th column of Stirling2 triangle A008277.

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A049435 Stirling numbers of second kind: 10th column of Stirling2 triangle A008277.

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A049447 Stirling numbers of second kind: 9th column of Stirling2 triangle A008277.

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A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets).

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A016094 Expansion of 1/((1-9x)(1-10x)(1-11x)(1-12*x)).

A016109 Expansion of 1/((1-7x)(1-8x)(1-9x)(1-10*x)).