A000770 -id:A000770 - cp's OEIS frontend

A000920 Differences of 0: 6!*Stirling2(n,6).

0, 0, 0, 0, 0, 720, 15120, 191520, 1905120, 16435440, 129230640, 953029440, 6711344640, 45674188560, 302899156560, 1969147121760, 12604139926560, 79694820748080, 499018753280880, 3100376804676480, 19141689213218880, 117579844328562000

Offset: 1

N. J. A. Sloane

Number of surjections from an n-element set onto a six-element set, with n >= 6. - Mohamed Bouhamida, Dec 15 2007

Number of rows of n colors using exactly six colors. For n=6, the 720 rows are the 720 permutations of ABCDEF. - Robert A. Russell, Sep 25 2018

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A001117, A001118, A000918, A000919, A000770.

Column 6 of A019538.

[6^n-Binomial(6,5)*5^n+Binomial(6,4)*4^n-Binomial(6,3)*3^n+Binomial(6,2)*2^n-Binomial(6,1): n in [1..30]]; // Vincenzo Librandi, May 18 2015

720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);

CoefficientList[Series[(720*x^5)/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)
k=6; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

a(n) = 6!*stirling(n, 6, 2); \\ Altug Alkan, Sep 25 2018

a(n) = Sum((-1)^i*binomial(6, i)*(6-i)^n, i = 0 .. 5).
a(n) = 6^n-C(6,5)*5^n+C(6,4)*4^n-C(6,3)*3^n+C(6,2)*2^n-C(6,1) with n>=6. - Mohamed Bouhamida, Dec 15 2007
G.f.: 720*x^6/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009]
a(n) = 720*A000770(n). - R. J. Mathar, Apr 30 2015
E.g.f.: (exp(x) - 1)^6. - Geoffrey Critzer, May 17 2015

A032180 Number of ways to partition n labeled elements into 6 pie slices.

Original entry on oeis.org

120, 2520, 31920, 317520, 2739240, 21538440, 158838240, 1118557440, 7612364760, 50483192760, 328191186960, 2100689987760, 13282470124680, 83169792213480, 516729467446080, 3190281535536480, 19596640721427000, 119876382958008600

Offset: 6

Christian G. Bower

For n>=6, a(n) is equal to the number of functions f: {1,2,...,n-1}->{1,2,3,4,5,6} such that Im(f) contains 5 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 6..1000
C. G. Bower, Transforms (2)
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A000770, A008277, A000225, A028243, A028244, A028245.

[5*2^(n-1)-10*3^(n-1)+10*4^(n-1)-5^n+6^(n-1)-1: n in [6..30]]; // Vincenzo Librandi, Oct 19 2013

with (combstruct):ZL:=[S, {S=Sequence(U, card=r), U=Set(Z, card>=1)}, labeled]: seq(count(subs(r=6, ZL), size=m)/6, m=6..21); # Zerinvary Lajos, Mar 08 2008

CoefficientList[Series[120/((x - 1) (2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 19 2013 *)
Table[120*StirlingS2[n,6], {n,6,30}] (* G. C. Greubel, Nov 19 2017 *)

for(n=6,30, print1(120*stirling(n,6,2), ", ")) \\ G. C. Greubel, Nov 19 2017

"CIJ[ 6 ]" (necklace, indistinct, labeled, 6 parts) transform of 1, 1, 1, 1...
a(n) = 120*S(n, 6).
From Emeric Deutsch, May 02 2004: (Start)
a(n) = 5*2^(n-1) - 10*3^(n-1) + 10*4^(n-1) - 5^n + 6^(n-1) - 1.
a(n) = 120*A000770(n). (End)
G.f.: 120*x^6/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Sep 03 2012
E.g.f.: (Sum_{k=0..6} (-1)^(6-k)*binomial(6,k)*exp(k*x))/6 with a(n) = 0 for n = 0..5. - Wolfdieter Lang, May 03 2017

More terms from Vincenzo Librandi Oct 19 2013

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

A056330 Number of reversible string structures with n beads using exactly six different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 12, 142, 1346, 11511, 89974, 662674, 4662574, 31724735, 210361046, 1367510326, 8752976610, 55343947975, 346541488998, 2153041587538, 13292844257198, 81652683550119, 499484958151630

Offset: 1

Marks R. Nester

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 six different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

			For a(7)=12, the color patterns are ABCDEFA, ABCDEBF, ABCDCEF, AABCDEF, ABACDEF, ABCADEF, ABCDAEF, ABBCDEF, ABCBDEF, ABCDBEF, and ABCCDEF. The first three 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 (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).

Column 6 of A284949.
Cf. A056313.
Cf. A000770 (oriented), A320529 (chiral), A304976 (achiral).

k=6; Table[(StirlingS2[n,k] + If[EvenQ[n], StirlingS2[n/2+3,6] - 3StirlingS2[n/2+2,6] - 8StirlingS2[n/2+1,6] + 16StirlingS2[n/2,6], 3StirlingS2[(n+5)/2,6] - 17StirlingS2[(n+3)/2,6] + 20StirlingS2[(n+1)/2,6]])/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 = 6; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 1, 12, 142, 1346, 11511, 89974, 662674, 4662574, 31724735}, 40] (* Robert A. Russell, Oct 14 2018 *)

a(n) = A056325(n) - A056324(n).
From Robert A. Russell, Oct 14 2018: (Start)
a(n) = (S2(n,k) + A(n,k))/2, where k=6 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^6 / Product_{k=1..6} (1 - k*x) + x^6 (1+x) (1-4x^2) (1+2x-x^2-4x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.
a(n) = (A000770(n) + A304976(n)) / 2 = A000770(n) - A320529(n) = A320529(n) + A304976(n). (End)

A320529 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638, 210332227, 1367416232, 8752773288, 55343303064, 346540112781, 2153037307846, 13292835205606, 81652655795106, 499484899831775, 3045117929546220, 18513208314957356, 112297592929814292, 679900657841661529, 4110073054119135194, 24814158520762637754

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(7)=9, the chiral pairs are AABCDEF-ABCDEFF, ABACDEF-ABCDEFE, ABCADEF-ABCDEFD, ABCDAEF-ABCDEFC, ABCDEAF-ABCDEFB, ABBCDEF-ABCDEEF, ABCBDEF-ABCDEDF, ABCDBEF-ABCDECF, and ABCCDEF-ABCDDEF.

Index entries for linear recurrences with constant coefficients, signature (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).

Column 6 of A320525.

Cf. A000770 (oriented), A056330 (unoriented), A304976 (achiral).

m:=30; R:=PowerSeriesRing(Integers(), m); [0,0,0,0,0,0] cat Coefficients(R!((x^6/(&*[1-k*x: k in [1..6]]) - x^6*(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/(&*[1-k*x^2: k in [1..6]]) )/2)); // G. C. Greubel, Oct 19 2018

k=6; Table[(StirlingS2[n,k] - If[EvenQ[n], StirlingS2[n/2+3,6] - 3StirlingS2[n/2+2,6] - 8StirlingS2[n/2+1,6] + 16StirlingS2[n/2,6], 3StirlingS2[(n+5)/2,6] - 17StirlingS2[(n+3)/2,6] + 20StirlingS2[(n+1)/2,6]])/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 = 6; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638}, 30]

x='x+O('x^30); concat(vector(6), Vec((x^6/prod(k=1,6, 1-k*x) - x^6* (1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/prod(k=1,6,(1-k*x^2)))/2)) \\ G. C. Greubel, Oct 19 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=6 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^6 / (Product_{k=1..6} (1 - k*x)) - x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.
a(n) = (A000770(n) - A304976(n)) / 2 = A000770(n) - A056330(n) = A056330(n) - A304976(n).

A056282 Number of primitive (aperiodic) word structures of length n which contain exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 21, 266, 2646, 22827, 179487, 1323651, 9321312, 63436352, 420693273, 2734926292, 17505749898, 110687248392, 693081601779, 4306078872557, 26585679462783, 163305339165738, 998969857983405

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]

Column 6 of A137651.

Cf. A056271.

a(n) = Sum_{n > 0, d|n} mu(d)*A000770(n/d).

G.f.: Sum_{k>=1} mu(k) * x^(6*k) / Product_{j=1..6} (1 - j*x^k). - Ilya Gutkovskiy, Apr 15 2021

A056475 Number of palindromic structures using exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 21, 21, 266, 266, 2646, 2646, 22827, 22827, 179487, 179487, 1323652, 1323652, 9321312, 9321312, 63436373, 63436373, 420693273, 420693273, 2734926558, 2734926558, 17505749898, 17505749898, 110687251039, 110687251039

Offset: 1

Marks R. Nester

Permuting the symbols will not change the 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]

Index entries for linear recurrences with constant coefficients, signature (1,20,-20,-155,155,580,-580,-1044,1044,720,-720).

Cf. A000770, A056471.

a(n) = stirling2( [(n+1)/2], 6).

G.f.: x^11/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)*(6*x^2-1)). [Colin Barker, Jul 24 2012]

A245602 Triangle read by rows: the negative terms of A163626.

Original entry on oeis.org

-1, -3, -7, -6, -15, -60, -31, -390, -120, -63, -2100, -2520, -127, -10206, -31920, -5040, -255, -46620, -317520, -181440, -511, -204630, -2739240, -3780000, -362880, -1023, -874500, -21538440, -59875200, -19958400, -2047

Offset: 0

Paul Curtz, Dec 17 2014

These numbers a(n) are the companion of A249163(n).
Consider the Worpitzky fractions A163626(n)/A002260(n) yielding the second Bernoulli numbers A164555(n)/A027642(n):
1,
1, -1/2,
1, -3/2,  +2/3,
1, -7/2, +12/3, -6/4,
etc.
From the second row on, the sum of the numerators is 0.
The absolute values of every row of the numerators triangle A163626 are 1, 2, 6, 26, ... = A000629(n).
a(n) triangle is shifted. It starts from second row and second column of triangle above.
-1,
-3,
-7,       -6,
-15,     -60,
-31,    -390,    -120,
-63,   -2100,   -2520,
-127, -10206,  -31920,   -5040,
-255, -46620, -317520, -181440,
etc.
Sum of successive rows: -1, -3, -13, -75, ... = -A000670(n+1).
Successive columns: A000225, A028244, from the Stirling numbers of second kind S(n,2), S(n,4), S(n,6), S(n,8), S(n,10), ... . See A000770, A032180, A049434, A228910, A049435, A228912, A008277.
Thanks to Jean-François Alcover.

Cf. A000225, A000629, A000670, A000770, A002260, A008277, A009445, A027642, A028244, A032180, A049434, A049435, A163626, A164555, A228910, A228912, A249163.

Select[ Table[ (-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten, Negative] (* Jean-François Alcover, Dec 26 2014 *)

cp's OEIS Frontend

A000920 Differences of 0: 6!*Stirling2(n,6).

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A032180 Number of ways to partition n labeled elements into 6 pie slices.

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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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A056330 Number of reversible string structures with n beads using exactly six different colors.

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A320529 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).

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