A007581 -id:A007581 - cp's OEIS frontend

A099263 a(n) = (1/40320)8^n + (1/1440)6^n + (1/360)5^n + (1/64)4^n + (11/180)3^n + (53/288)2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).

1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115929, 677359, 4189550, 27243100, 184941915, 1301576801, 9433737120, 69998462014, 529007272061, 4054799902003, 31415584940850, 245382167055488, 1928337630016767, 15222915798289765, 120582710957928740, 957566218595705122, 7618489083072350433

Offset: 1

Nelma Moreira, Oct 10 2004

Density of regular language L over {1,2,3,4,5,6,7,8} (i.e., number of strings of length n in L) described by a regular expression with c = 8: Sum_{i=1..c} Product_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
N. Moreira and R. Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Index entries for linear recurrences with constant coefficients, signature (29,-343,2135,-7504,14756,-14832,5760).

Cf. A007051, A007581, A056272, A056273, A099262.
A row of the array in A278984.
Cf. A008277 (Stirling2), A248925.

[(1/40320)*8^n+(1/1440)*6^n+(1/360)*5^n+(1/64)*4^n +(11/180)*3^n+(53/288)*2^n+103/280: n in [1..30]]; // Vincenzo Librandi, Jul 27 2017

CoefficientList[Series[-(3641 x^6 - 6583 x^5 + 4566 x^4 - 1579 x^3 + 290 x^2 - 27 x + 1) / ((x - 1) (2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1) (8 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 27 2017 *)
Table[Sum[StirlingS2[n, k], {k, 0, 8}], {n, 1, 30}] (* Robert A. Russell, Apr 25 2018 *)
LinearRecurrence[{29,-343,2135,-7504,14756,-14832,5760},{1,2,5,15,52,203,877},30] (* Harvey P. Dale, Aug 27 2019 *)

a(n) = (1/40320)*8^n + (1/1440)*6^n + (1/360)*5^n + (1/64)*4^n + (11/180)*3^n + (53/288)*2^n + 103/280; \\ Altug Alkan, Apr 25 2018

For c = 8, a(n) = c^n/c! + Sum_{k=1..c-2} k^n/k! * Sum_{j=2..c-k} (-1)^j/j!, or = Sum_{k=1..c} g(k, c)*k^n, where g(1, 1) = 1, g(1, c) = g(1, c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k, c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.
G.f.: -x*(3641*x^6 - 6583*x^5 + 4566*x^4 - 1579*x^3 + 290*x^2 - 27*x + 1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). [Colin Barker, Dec 05 2012]
a(n) = Sum_{k=0..8} Stirling2(n,k).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} (1 - j*x) with k = 8. - Robert A. Russell, Apr 25 2018

More terms from Michel Marcus, Jan 05 2025

A208201 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to more than two of its immediate leftward or upward or left-upward diagonal neighbors.

Original entry on oeis.org

1, 2, 2, 5, 14, 5, 15, 178, 178, 15, 51, 2653, 10366, 2653, 51, 187, 41272, 639701, 639701, 41272, 187, 715, 648569, 39725054, 156617480, 39725054, 648569, 715, 2795, 10215440, 2468504665, 38372808893, 38372808893, 2468504665, 10215440, 2795

Offset: 1

R. H. Hardin Feb 24 2012

Table starts
....1.........2.............5.................15......................51
....2........14...........178...............2653...................41272
....5.......178.........10366.............639701................39725054
...15......2653........639701..........156617480.............38372808893
...51.....41272......39725054........38372808893..........37069738362057
..187....648569....2468504665......9402035642924.......35810971705853725
..715..10215440..153402542384...2303673397508915....34594950233928279876
.2795.160984657.9533100532733.564442821781056268.33420220819293699707599

			Some solutions for n=4 k=3
..0..0..1....0..0..0....0..1..1....0..0..0....0..1..1....0..0..0....0..1..0
..0..2..0....0..1..1....0..0..0....1..1..0....1..0..0....0..1..2....0..2..3
..1..2..1....2..0..2....0..1..1....0..1..0....0..1..0....1..1..3....3..0..0
..3..0..0....2..1..0....0..2..1....0..1..0....1..0..0....1..0..0....2..1..3

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A007581(n-1)

A208864 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to more than one of its immediate leftward or upward or left-upward diagonal neighbors.

Original entry on oeis.org

1, 2, 2, 5, 11, 5, 15, 127, 127, 15, 51, 1691, 5796, 1691, 51, 187, 23047, 273049, 273049, 23047, 187, 715, 315203, 12883280, 44452082, 12883280, 315203, 715, 2795, 4313071, 607924387, 7240366702, 7240366702, 607924387, 4313071, 2795, 11051, 59022155

Offset: 1

R. H. Hardin Mar 02 2012

Table starts
....1........2.............5................15....................51
....2.......11...........127..............1691.................23047
....5......127..........5796............273049..............12883280
...15.....1691........273049..........44452082............7240366702
...51....23047......12883280........7240366702.........4071491223248
..187...315203.....607924387.....1179377179523......2289701830963879
..715..4313071...28686344276...192108983085145...1287681340443915644
.2795.59022155.1353633375649.31292693945057258.724166398812363542260

			Some solutions for n=4 k=3
..0..0..1....0..0..0....0..1..1....0..1..0....0..0..0....0..0..0....0..0..0
..0..2..0....0..1..2....1..0..2....0..2..3....0..1..2....0..1..1....0..1..1
..1..2..1....1..2..3....3..0..1....3..0..0....2..1..2....0..1..2....1..2..0
..3..0..0....1..0..0....1..1..2....2..1..3....0..1..2....2..3..2....1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..144

Column 1 is A007581(n-1)

A223062 T(n,k)=Number of nXk 0..3 arrays with all horizontally or vertically connected equal values in a straight line, and new values 0..3 introduced in row major order.

Original entry on oeis.org

1, 2, 2, 5, 10, 5, 15, 114, 114, 15, 51, 1473, 4865, 1473, 51, 187, 19360, 213469, 213469, 19360, 187, 715, 255045, 9383963, 31331820, 9383963, 255045, 715, 2795, 3361024, 412602889, 4606719559, 4606719559, 412602889, 3361024, 2795, 11051, 44294253

Offset: 1

R. H. Hardin Mar 13 2013

Table starts
...1......2.........5...........15...............51.................187
...2.....10.......114.........1473............19360..............255045
...5....114......4865.......213469..........9383963...........412602889
..15...1473....213469.....31331820.......4606719559........677531592132
..51..19360...9383963...4606719559....2266567594841....1115636153544055
.187.255045.412602889.677531592132.1115636153544055.1837966217755803448

			Some solutions for n=3 k=4
..0..1..0..1....0..1..0..2....0..1..0..2....0..1..0..1....0..1..1..0
..2..2..0..1....2..1..0..2....2..3..0..2....2..1..2..2....0..2..3..0
..3..3..3..3....2..3..1..0....2..1..2..0....3..2..3..0....0..2..0..1

R. H. Hardin, Table of n, a(n) for n = 1..127

Column 1 is A007581(n-1)

Empirical for column k:
k=1: a(n) = 7*a(n-1) -14*a(n-2) +8*a(n-3)
k=2: a(n) = 18*a(n-1) -71*a(n-2) +102*a(n-3) -48*a(n-4) for n>5
k=3: a(n) = 55*a(n-1) -515*a(n-2) +1337*a(n-3) -696*a(n-4) -1188*a(n-5) +1008*a(n-6) for n>7
k=4: [order 11 for n>12]

A206396 T(n,k)=Number of nXk 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 4, 2, 5, 25, 26, 25, 5, 15, 172, 206, 206, 172, 15, 51, 1201, 1592, 1931, 1592, 1201, 51, 187, 8404, 12428, 16784, 16784, 12428, 8404, 187, 715, 58825, 96632, 151630, 170796, 151630, 96632, 58825, 715, 2795, 411772, 752552, 1343560

Offset: 1

R. H. Hardin Feb 07 2012

Table starts
...1.....1......1........2.........5.........15..........51..........187
...1.....1......4.......25.......172.......1201........8404........58825
...1.....4.....26......206......1592......12428.......96632.......752552
...2....25....206.....1931.....16784.....151630.....1343560.....12046648
...5...172...1592....16784....170796....1787258....18574298....193499878
..15..1201..12428...151630...1787258...21983256...268956972...3301485294
..51..8404..96632..1343560..18574298..268956972..3889732730..56960076094
.187.58825.752552.12046648.193499878.3301485294.56960076094.998388746378

			Some solutions for n=4 k=3
..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
..2..3..0....2..3..4....2..3..4....3..4..5....3..4..5....2..3..0....2..3..4
..1..4..5....1..0..5....4..5..1....5..0..1....1..0..3....4..5..1....1..5..0
..0..2..1....3..2..1....1..2..0....4..3..2....2..5..1....0..2..3....0..2..3

R. H. Hardin, Table of n, a(n) for n = 1..127

Column 1 is A007581(n-3)

Column 2 is A034494(n-2)

A208175 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to more than one of its immediate leftward or upward neighbors.

Original entry on oeis.org

1, 2, 2, 5, 13, 5, 15, 159, 159, 15, 51, 2277, 8469, 2277, 51, 187, 33831, 473928, 473928, 33831, 187, 715, 506493, 26625693, 99752346, 26625693, 506493, 715, 2795, 7594479, 1496307201, 21002874867, 21002874867, 1496307201, 7594479, 2795, 11051

Offset: 1

R. H. Hardin Feb 24 2012

Table starts
....1.........2.............5.................15.....................51
....2........13...........159...............2277..................33831
....5.......159..........8469.............473928...............26625693
...15......2277........473928...........99752346............21002874867
...51.....33831......26625693........21002874867.........16567902881244
..187....506493....1496307201......4422202024212......13069424064243420
..715...7594479...84091222866....931104734002155...10309684155118737894
.2795.113908437.4725865313907.196046229442844736.8132690993759302714128

			Some solutions for n=4 k=3
..0..0..0....0..0..0....0..1..0....0..1..0....0..1..1....0..0..0....0..0..1
..0..1..2....0..1..0....0..2..2....0..2..3....1..0..0....0..1..1....0..1..0
..1..2..3....1..2..1....1..2..1....3..0..0....0..1..0....1..2..0....0..0..1
..1..0..0....0..1..3....1..3..2....2..1..3....1..0..1....1..0..1....1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A007581(n-1)

Column 2 is A205163(n-1)

A060919 Number of corners in a 4-sided fractal.

Original entry on oeis.org

4, 8, 20, 60, 204, 748, 2860, 11180, 44204, 175788, 701100, 2800300, 11193004, 44755628, 178989740, 715893420, 2863442604, 11453508268, 45813508780, 183252986540, 733009849004, 2932035201708, 11728132418220, 46912512895660, 187650018028204, 750600005003948, 3002399885798060

Offset: 1

Henry Bottomley, Apr 10 2001

Harry J. Smith, Table of n, a(n) for n=1,...,200
Henry Bottomley, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007581, A002450, A028400.

LinearRecurrence[{7,-14,8},{4,8,20},30] (* Harvey P. Dale, Sep 01 2023 *)

a(n) = { (2^n + 2)*(2^n + 4)/6 } \\ Harry J. Smith, Jul 14 2009

a(n) = 4^n/6+2^n+4/3 = (2^n+2)*(2^n+4)/6 = 4*A007581(n-1) = 4(a(n-1)-1)-2^n = A028400(n-1)-A002450(n-1).
From Colin Barker, Nov 28 2012: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3).
G.f.: -4*x*(5*x^2-5*x+1) / ((x-1)*(2*x-1)*(4*x-1)). (End)
E.g.f.: (exp(4*x) + 6*exp(2*x) + 8*exp(x) - 15)/6. - Stefano Spezia, Dec 26 2024

A164863 Number of ways of placing n labeled balls into 9 indistinguishable boxes; word structures of length n using a 9-ary alphabet.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678514, 4211825, 27602602, 190077045, 1368705291, 10254521370, 79527284317, 635182667816, 5199414528808, 43426867585575, 368654643520692, 3170300933550687, 27542984610086665, 241205285284001240

Offset: 0

Alois P. Heinz, Aug 28 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Moreira, N.; Reis, R. "On the Density of Languages Representing Finite Set Partitions", Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Eric Weisstein's World of Mathematics, Set Partition
Index entries for linear recurrences with constant coefficients, signature (37, -574, 4858, -24409, 74053, -131256, 122652, -45360).

Cf. A000110, A048993, A008291, A098825, A000012, A000079, A007051, A007581, A124303, A056272, A056273, A099262, A099263, A164864.

A row of the array in A278984.

# first program:
a:= n-> ceil(103/560*2^n +53/864*3^n +11/720*4^n +5^n/320 +6^n/2160 +7^n/10080 +9^n/362880): seq(a(n), n=0..25);
# second program:
a:= n-> add(Stirling2(n, k), k=0..9): seq(a(n), n=0..25);

Table[Sum[StirlingS2[n, k], {k, 0, 9}], {n, 0, 30}] (* Robert A. Russell, Apr 25 2018 *)

a(n) = Sum_{k=0..9} stirling2 (n,k).
a(n) = ceiling (103/560*2^n +53/864*3^n +11/720*4^n +5^n/320 +6^n/2160 +7^n/10080 +9^n/362880).
G.f.: (16687*x^8 -67113*x^7 +88620*x^6 -56993*x^5 +20529*x^4 -4353*x^3 +539*x^2 -36*x+1) / ((9*x-1) *(7*x-1) *(6*x-1) *(5*x-1) *(4*x-1) *(3*x-1) *(2*x-1) *(x-1)).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=9. - Robert A. Russell, Apr 25 2018

A223041 T(n,k)=Number of nXk 0..3 arrays with no more than floor(nXk/2) elements equal to at least one horizontal, diagonal or antidiagonal neighbor, with new values introduced in row major 0..3 order.

Original entry on oeis.org

1, 1, 2, 2, 8, 5, 11, 98, 105, 15, 34, 1265, 3780, 1585, 51, 131, 17356, 229437, 255943, 23107, 187, 438, 241583, 8580619, 42964721, 10904199, 353631, 715, 2150, 3395532, 545607733, 7361951209, 8206670940, 751990800, 5338646, 2795, 7676, 48048796

Offset: 1

R. H. Hardin Mar 12 2013

Table starts
.....1........1...........2............11............34...........131
.....2........8..........98..........1265.........17356........241583
.....5......105........3780........229437.......8580619.....545607733
....15.....1585......255943......42964721....7361951209.1278533695058
....51....23107....10904199....8206670940.3829674402996
...187...353631...751990800.1590576924149
...715..5338646.33007705273
..2795.81874696
.11051

			Some solutions for n=3 k=4
..0..1..0..2....0..1..2..3....0..1..0..0....0..1..2..0....0..1..2..2
..3..1..3..2....0..1..1..2....2..1..3..2....3..0..2..3....0..1..2..3
..2..2..1..1....2..3..0..3....3..2..0..2....2..0..1..2....3..1..2..3

R. H. Hardin, Table of n, a(n) for n = 1..49

Column 1 is A007581(n-1)
Column 2 is A222839
Row 1 is A222650

A241119 T(n,k)=Number of nXk 0..3 arrays with no element equal to exactly three horizontal or vertical neighbors, with new values 0..3 introduced in row major order.

Original entry on oeis.org

1, 2, 2, 5, 15, 5, 15, 178, 178, 15, 51, 2614, 9918, 2614, 51, 187, 40148, 587555, 587555, 40148, 187, 715, 622645, 35000157, 134229632, 35000157, 622645, 715, 2795, 9676364, 2085879115, 30679522712, 30679522712, 2085879115, 9676364, 2795

Offset: 1

R. H. Hardin, Apr 16 2014

Table starts
...1......2..........5............15................51..................187
...2.....15........178..........2614.............40148...............622645
...5....178.......9918........587555..........35000157...........2085879115
..15...2614.....587555.....134229632.......30679522712........7012241396116
..51..40148...35000157...30679522712....26891788727245....23571710809729613
.187.622645.2085879115.7012241396116.23571710809729613.79236581561126536192

			Some solutions for n=3 k=4
..0..1..1..0....0..1..1..0....0..1..0..1....0..1..1..0....0..0..1..1
..2..1..0..2....0..1..2..1....2..0..3..3....2..0..0..2....2..0..0..2
..0..0..3..0....3..2..0..3....3..1..1..1....3..2..2..3....1..1..1..3

R. H. Hardin, Table of n, a(n) for n = 1..84

Column 1 is A007581(n-1)

Column 2 is A200795

Empirical for column k:
k=1: a(n) = 7*a(n-1) -14*a(n-2) +8*a(n-3)
k=2: [order 8] for n>10
k=3: [order 20] for n>21
k=4: [order 74] for n>75

cp's OEIS Frontend

A099263 a(n) = (1/40320)*8^n + (1/1440)*6^n + (1/360)*5^n + (1/64)*4^n + (11/180)*3^n + (53/288)*2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).

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Magma

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A208201 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to more than two of its immediate leftward or upward or left-upward diagonal neighbors.

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A208864 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to more than one of its immediate leftward or upward or left-upward diagonal neighbors.

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A223062 T(n,k)=Number of nXk 0..3 arrays with all horizontally or vertically connected equal values in a straight line, and new values 0..3 introduced in row major order.

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Crossrefs

Formula

A206396 T(n,k)=Number of nXk 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order.

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Comments

Examples

Links

Crossrefs

A208175 T(n,k)=Number of nXk 0..3 arrays with new values 0..3 introduced in row major order and no element equal to more than one of its immediate leftward or upward neighbors.

Views

Author

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Comments

Examples

Links

Crossrefs

A060919 Number of corners in a 4-sided fractal.

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Links

Crossrefs

Programs

Mathematica

PARI

Formula

A164863 Number of ways of placing n labeled balls into 9 indistinguishable boxes; word structures of length n using a 9-ary alphabet.

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Programs

Maple

Mathematica

Formula

A223041 T(n,k)=Number of nXk 0..3 arrays with no more than floor(nXk/2) elements equal to at least one horizontal, diagonal or antidiagonal neighbor, with new values introduced in row major 0..3 order.

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A099263 a(n) = (1/40320)8^n + (1/1440)6^n + (1/360)5^n + (1/64)4^n + (11/180)3^n + (53/288)2^n + 103/280. Partial sum of Stirling numbers of second kind S(n,i), i=1..8 (i.e., a(n) = Sum_{i=1..8} S(n,i)).