A007581 -id:A007581 - cp's OEIS frontend

A164864 Number of ways of placing n labeled balls into 10 indistinguishable boxes; word structures of length n using a 10-ary alphabet.

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678569, 4213530, 27641927, 190829797, 1381367941, 10448276360, 82285618467, 672294831619, 5676711562593, 49344452550230, 439841775811967, 4005444732928641, 37136385907400125, 349459367068932740

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"
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3-4, 16-18.
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.
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Eric Weisstein's World of Mathematics, Set Partition
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (46,-906,9996,-67809,291774,-790964,1290824,-1136160,403200).

Cf. A000110, A048993, A008291, A098825, A000012, A000079, A007051, A007581, A124303, A056272, A056273, A099262, A099263, A164863.

A row of the array in A278984.

# First program:
a:= n-> ceil(2119/11520*2^n +103/1680*3^n +53/3456*4^n +11/3600*5^n +6^n/1920 +7^n/15120 +8^n/80640 +10^n/3628800): seq(a(n), n=0..25);
# second program:
a:= n-> add(Stirling2(n, k), k=0..10): seq(a(n), n=0..25);

Table[Sum[StirlingS2[n,k],{k,0,10}],{n,0,30}] (* Harvey P. Dale, Nov 22 2023 *)

a(n) = Sum_{k=0..10} Stirling2 (n,k).
a(n) = ceiling(2119/11520*2^n +103/1680*3^n +53/3456*4^n +11/3600*5^n +6^n/1920 +7^n/15120 +8^n/80640 +10^n/3628800).
G.f.: (148329*x^9 -613453*x^8 +855652*x^7 -596229*x^6 +240065*x^5 -59410*x^4 +9177*x^3 -862*x^2 +45*x-1) / ((10*x-1) *(8*x-1) *(7*x-1) *(6*x-1) *(5*x-1) *(4*x-1) *(3*x-1) *(2*x-1) *(x-1)).
a(n) <= A000110(n) with equality only for n <= 10.

A124303 Number of set partitions of length <= 4; sum of first 4 columns of triangle of Stirling numbers of 2nd kind; dimension of space of symmetric polynomials in 4 noncommuting variables.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 187, 715, 2795, 11051, 43947, 175275, 700075, 2798251, 11188907, 44747435, 178973355, 715860651, 2863377067, 11453377195, 45813246635, 183252462251, 733008800427, 2932033104555, 11728128223915, 46912504507051, 187650001250987

Offset: 0

Mike Zabrocki, Oct 25 2006

Apart from initial term, same as A007581. - Valery A. Liskovets, Nov 16 2006

			Number of set partitions of {1,2,3,4,5,6} are given by A008277(6,k) = 1, 31, 90, 65, 15, 1 and hence a(6) = 1+31+90+65 = 187.

Colin Barker, Table of n, a(n) for n = 0..1000
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables, Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A124292, A000110, A008277.

A row of the array in A278984.

a:=proc(n); if n<4 then [1,1,2,5][n+1]; else 7*a(n-1)-14*a(n-2)+8*a(n-3); fi; end:

Join[{1}, LinearRecurrence[{7, -14, 8}, {1, 2, 5}, 26]] (* Jean-François Alcover, Nov 20 2017 *)
Table[Sum[StirlingS2[n,k],{k,0,4}],{n,0,40}] (* Robert A. Russell, Mar 29 2018 *)

Vec((1 - 6*x + 9*x^2 - 3*x^3) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Nov 03 2017

O.g.f.: (3*q^3 - 9*q^2 + 6*q - 1)/(8*q^3 - 14*q^2 + 7*q - 1) = Sum_{k=0..4} (q^k/Product_{i=1..k} (1-i*q)).
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3); a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 5, a(n) = Sum_{k=1..4} A008277(n,k).
a(n) = (8 + 3*2^(1+n) + 4^n) / 24 for n>0. - Colin Barker, Nov 03 2017
a(n) = Sum_{k=0..4} Stirling2(n,k). - Robert A. Russell, Mar 29 2018
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=4. - Robert A. Russell, Apr 25 2018
E.g.f.: (9 + 8*exp(x) + 6*exp(2*x) + exp(4*x))/24. - Peter Luschny, Nov 06 2018

A241328 T(n,k)=Number of nXk 0..3 arrays with no element equal to fewer vertical neighbors than horizontal neighbors, with new values 0..3 introduced in row major order.

Original entry on oeis.org

1, 1, 2, 2, 8, 5, 5, 59, 85, 15, 14, 530, 2344, 1030, 51, 41, 4877, 68935, 95144, 13011, 187, 122, 45057, 2034543, 8949808, 3875244, 165924, 715, 365, 416533, 60066019, 842185933, 1162535788, 157912026, 2121033, 2795, 1094, 3851085, 1773370241

Offset: 1

R. H. Hardin, Apr 19 2014

Table starts
...1.......1..........2..............5................14....................41
...2.......8.........59............530..............4877.................45057
...5......85.......2344..........68935...........2034543..............60066019
..15....1030......95144........8949808.........842185933...........79254376889
..51...13011....3875244.....1162535788......348722168314.......104609549355169
.187..165924..157912026...151022716125...144410575985227....138095118530911728
.715.2121033.6435036610.19619065042282.59802363257355728.182299462522915741748

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

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A007581(n-1)

Row 1 is A007051(n-2)

Empirical for column k:
k=1: a(n) = 7*a(n-1) -14*a(n-2) +8*a(n-3)
k=2: [order 9] for n>10
k=3: [order 15]
k=4: [order 53]
Empirical for row n:
n=1: a(n) = 4*a(n-1) -3*a(n-2) for n>3
n=2: a(n) = 12*a(n-1) -26*a(n-2) +2*a(n-3) +28*a(n-4) -6*a(n-5) -9*a(n-6) for n>8
n=3: [order 11] for n>13
n=4: [order 36] for n>38

A222878 T(n,k)=Number of nXk 0..3 arrays with no more than floor(nXk/2) elements equal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..3 order.

Original entry on oeis.org

1, 1, 2, 2, 11, 5, 11, 122, 131, 15, 34, 1814, 6282, 2150, 51, 131, 26311, 409775, 443510, 32491, 187, 438, 399849, 19707855, 95867967, 23794078, 533186, 715, 2150, 6013669, 1340197778, 20978414941, 22794825314, 1698174722, 8362152, 2795, 7676

Offset: 1

R. H. Hardin Mar 08 2013

Table starts
.....1..........1...............2...................11......................34
.....2.........11.............122.................1814...................26311
.....5........131............6282...............409775................19707855
....15.......2150..........443510.............95867967.............20978414941
....51......32491........23794078..........22794825314..........16473931317022
...187.....533186......1698174722........5472896849943.......17791003053622809
...715....8362152.....93656293632.....1322676376672081....14337592109053271558
..2795..135650378...6637413406814...321195816580955511.15577784123509878927189
.11051.2155759835.372075279639542.78284744747263499257

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

R. H. Hardin, Table of n, a(n) for n = 1..82

Column 1 is A007581(n-1)

Row 1 is A222650

A028401 The (2^n+1)-th triangular number (cf. A000217).

Original entry on oeis.org

3, 6, 15, 45, 153, 561, 2145, 8385, 33153, 131841, 525825, 2100225, 8394753, 33566721, 134242305, 536920065, 2147581953, 8590131201, 34360131585, 137439739905, 549757386753, 2199026401281, 8796099313665, 35184384671745

Offset: 2

N. J. A. Sloane

Number of types of Boolean functions of n variables under a certain group.

Also the number of ordered decompositions of 2^n into 3 nonnegative integers (e.g., 2 = 0+0+2 = 0+2+0 = 2+0+0 = 1+1+0 = 1+0+1 = 0+1+1). - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Daniel Poveda Parrilla, Illustration of initial terms
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Equals 2*A036562(n-4) - 1, n > 3.

Cf. A000217.

Drop[#, 2] &@ CoefficientList[Series[3 x^2*(1 - 5 x + 5 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Jul 08 2019 *)

def A028401(n): return ((m:=1<2 else 3 # Chai Wah Wu, Jul 11 2024

From Ralf Stephan, Aug 23 2003: (Start)
a(n) = (3/8)*2^n + (1/32)*4^n + 1.
a(n) = 3*A007581(n-2) = (3/4)*A060919(n-1). (End)
a(n) = (2^n+4)*(2^n+8)/32. - Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007
G.f.: 3*x^2*(1-5*x+5*x^2)/((1-x)*(1-2*x)*(1-4*x)). - Colin Barker, Mar 09 2012
a(n) = a(n-1) + 3*A000217(2^(n-3)) for n > 2. - Daniel Poveda Parrilla, Dec 27 2016
E.g.f.: (32*exp(x) + 12*exp(2*x) + exp(4*x) - 45 - 60*x)/32. - Stefano Spezia, Jul 11 2024

More terms from Vladeta Jovovic, Feb 24 2000

Simpler definition from Tamas Kalmar-Nagy (integers(AT)kalmarnagy.com), Aug 02 2007

A200801 T(n,k) is the number of n X k 0..3 arrays with values 0..3 introduced in row major order and each element equal to no more than two horizontal and vertical neighbors.

Original entry on oeis.org

1, 2, 2, 5, 15, 5, 15, 178, 178, 15, 51, 2614, 9880, 2614, 51, 187, 40148, 583813, 583813, 40148, 187, 715, 622645, 34679839, 132636590, 34679839, 622645, 715, 2795, 9676364, 2060918000, 30147154218, 30147154218, 2060918000, 9676364, 2795

Offset: 1

R. H. Hardin, Nov 22 2011

			Table starts:
.....1...........2.................5......................15
.....2..........15...............178....................2614
.....5.........178..............9880..................583813
....15........2614............583813...............132636590
....51.......40148..........34679839.............30147154218
...187......622645........2060918000...........6852264918471
...715.....9676364......122478253815........1557479347400065
..2795...150442627.....7278777317468......354005859128023982
.11051..2339207390...432571571252989....80463441477635545163
.43947.36372631268.25707362563355693.18288865135614195620421
...
Some solutions for n=5 and k=3:
..0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0
..2..0..0....0..1..1....0..0..2....2..2..0....2..2..0....2..2..0....2..1..2
..2..3..3....1..0..2....3..1..0....1..1..1....3..2..2....0..0..1....0..1..3
..2..2..0....3..2..1....3..3..0....2..2..1....3..1..3....1..1..1....1..1..2

Ron Hardin, Table of n, a(n) for n = 1..97

Main diagonal is A200794.
Columns 1..7 are A007581(n-1), A200795, A200796, A200797, A200798, A200799, A200780.
Cf. A199655, A198528.

A203647 T(n,k) = number of arrays of n 0..k integers with new values introduced in order 0..k but otherwise unconstrained. Array read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 5, 8, 1, 2, 5, 14, 16, 1, 2, 5, 15, 41, 32, 1, 2, 5, 15, 51, 122, 64, 1, 2, 5, 15, 52, 187, 365, 128, 1, 2, 5, 15, 52, 202, 715, 1094, 256, 1, 2, 5, 15, 52, 203, 855, 2795, 3281, 512, 1, 2, 5, 15, 52, 203, 876, 3845, 11051, 9842, 1024, 1, 2, 5, 15, 52, 203, 877

Offset: 1

R. H. Hardin, Jan 04 2012

Table starts
....1.....1......1......1......1......1......1......1......1......1......1
....2.....2......2......2......2......2......2......2......2......2......2
....4.....5......5......5......5......5......5......5......5......5......5
....8....14.....15.....15.....15.....15.....15.....15.....15.....15.....15
...16....41.....51.....52.....52.....52.....52.....52.....52.....52.....52
...32...122....187....202....203....203....203....203....203....203....203
...64...365....715....855....876....877....877....877....877....877....877
..128..1094...2795...3845...4111...4139...4140...4140...4140...4140...4140
..256..3281..11051..18002..20648..21110..21146..21147..21147..21147..21147
..512..9842..43947..86472.109299.115179.115929.115974.115975.115975.115975
.1024.29525.175275.422005.601492.665479.677359.678514.678569.678570.678570
Lower left triangular part seems to be A102661. - R. J. Mathar, Nov 29 2015

			Some solutions for n=7, k=5:
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....1....1....1....1....0....0....1....1....1....1....1....1....1....1
..1....0....2....1....2....2....1....1....2....2....2....2....1....2....1....2
..0....1....1....0....3....3....2....2....1....3....1....1....1....0....0....2
..0....0....3....1....0....4....3....0....2....3....1....1....1....0....2....1
..2....2....4....2....2....0....4....2....0....2....2....3....2....3....2....0
..1....3....1....0....2....5....0....0....0....0....0....2....2....1....1....1

R. H. Hardin, Table of n, a(n) for n = 1..10011

Column 1 is A000079(n-1).
Column 2 is A007051(n-1).
Column 3 is A007581(n-1).
Column 4 is A056272.
Column 5 is A056273.
Column 6 is A099262.
Column 7 is A099263.
Column 8 is A164863.
Column 9 is A164864.
Column 10 is A203641.
Column 11 is A203642.
Column 12 is A203643.
Column 13 is A203644.
Column 14 is A203645.
Column 15 is A203646.
Diagonal is A000110.

T:= proc(n,k) option remember;  if k = 1 then 2^(n-1)
else 1 + add(binomial(n-1,j-1)*procname(n-j,k-1),j=1..n-1)
fi
end proc:
seq(seq(T(k,m-k),k=1..m-1),m=2..10); # Robert Israel, May 20 2016

T[n_, k_] := Sum[StirlingS2[n, j], {j, 1, k+1}]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)

T(n,k) = Sum_{j = 1..k+1} Stirling2(n,j). - Andrew Howroyd, Mar 19 2017
T(n,k) = A278984(k+1, n). - Andrew Howroyd, Mar 19 2017
Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 4*a(n-1) -3*a(n-2)
k=3: a(n) = 7*a(n-1) -14*a(n-2) +8*a(n-3)
k=4: a(n) = 11*a(n-1) -41*a(n-2) +61*a(n-3) -30*a(n-4)
k=5: a(n) = 16*a(n-1) -95*a(n-2) +260*a(n-3) -324*a(n-4) +144*a(n-5)
k=6: a(n) = 22*a(n-1) -190*a(n-2) +820*a(n-3) -1849*a(n-4) +2038*a(n-5) -840*a(n-6)
k=7: a(n) = 29*a(n-1) -343*a(n-2) +2135*a(n-3) -7504*a(n-4) +14756*a(n-5) -14832*a(n-6) +5760*a(n-7)
k=8: a(n) = 37*a(n-1) -574*a(n-2) +4858*a(n-3) -24409*a(n-4) +74053*a(n-5) -131256*a(n-6) +122652*a(n-7) -45360*a(n-8)
k=9: a(n) = 46*a(n-1) -906*a(n-2) +9996*a(n-3) -67809*a(n-4) +291774*a(n-5) -790964*a(n-6) +1290824*a(n-7) -1136160*a(n-8) +403200*a(n-9)
k=10: a(n) = 56*a(n-1) -1365*a(n-2) +19020*a(n-3) -167223*a(n-4) +965328*a(n-5) -3686255*a(n-6) +9133180*a(n-7) -13926276*a(n-8) +11655216*a(n-9) -3991680*a(n-10)
k=11: a(n) = 67*a(n-1) -1980*a(n-2) +33990*a(n-3) -375573*a(n-4) +2795331*a(n-5) -14241590*a(n-6) +49412660*a(n-7) -113667576*a(n-8) +163671552*a(n-9) -131172480*a(n-10) +43545600*a(n-11)
k=12: a(n) = 79*a(n-1) -2783*a(n-2) +57695*a(n-3) -782133*a(n-4) +7284057*a(n-5) -47627789*a(n-6) +219409685*a(n-7) -703202566*a(n-8) +1519272964*a(n-9) -2082477528*a(n-10) +1606986720*a(n-11) -518918400*a(n-12)
k=13: a(n) = 92*a(n-1) -3809*a(n-2) +93808*a(n-3) -1530243*a(n-4) +17419116*a(n-5) -141963107*a(n-6) +835933384*a(n-7) -3542188936*a(n-8) +10614910592*a(n-9) -21727767984*a(n-10) +28528276608*a(n-11) -21289201920*a(n-12) +6706022400*a(n-13)
k=14: a(n) = 106*a(n-1) -5096*a(n-2) +147056*a(n-3) -2840838*a(n-4) +38786748*a(n-5) -385081268*a(n-6) +2816490248*a(n-7) -15200266081*a(n-8) +59999485546*a(n-9) -169679309436*a(n-10) +331303013496*a(n-11) -418753514880*a(n-12) +303268406400*a(n-13) -93405312000*a(n-14)
k=15: a(n) = 121*a(n-1) -6685*a(n-2) +223405*a(n-3) -5042947*a(n-4) +81308227*a(n-5) -965408015*a(n-6) +8576039615*a(n-7) -57312583328*a(n-8) +287212533608*a(n-9) -1066335473840*a(n-10) +2866534951280*a(n-11) -5367984964224*a(n-12) +6557974412544*a(n-13) -4622628648960*a(n-14) +1394852659200*a(n-15)
From Robert Israel, May 20 2016: (Start)
T(n,k) = 1 + Sum_{j=1..n-1} binomial(n-1,j-1)*T(n-j,k-1).
G.f. for columns g_k(z) satisfies g_k(z) = (z/(1-z))*(1+ g_{k-1}(z/(1-z))) with g_1(z) = z/(1-2z).
Thus g_k is a rational function: it has a simple pole at z=1/j for 1<=j<=k+1 except j=k, and it has a finite limit at infinity (so the degree of the numerator is k).  This implies that column k satisfies the recurrences listed above, whose coefficients correspond to the expansion of (z-1/(k+1))* Product_{j=1..k-1}(z - 1/j).
(End)

A208353 T(n,k) is the number of n X k 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 2, 2, 5, 15, 5, 15, 100, 100, 15, 51, 868, 1095, 868, 51, 187, 7780, 12625, 12625, 7780, 187, 715, 69988, 153237, 230387, 153237, 69988, 715, 2795, 629860, 1901508, 4773885, 4773885, 1901508, 629860, 2795, 11051, 5668708, 23658861, 103672036

Offset: 1

R. H. Hardin, Feb 25 2012

			Table starts:
....1.......2.........5..........15.............51............187
....2......15.......100.........868...........7780..........69988
....5.....100......1095.......12625.........153237........1901508
...15.....868.....12625......230387........4773885......103672036
...51....7780....153237.....4773885......191586797.....8045978096
..187...69988...1901508...103672036.....8045978096...647640659639
..715..629860..23658861..2280287753...340596199800.52428246114853
.2795.5668708.294608660.50481169071.14513602070899
Some solutions for n=4 and k=3:
..0..0..0....0..0..0....0..0..0....0..0..0....0..1..0....0..0..0....0..0..0
..1..0..1....1..1..1....1..2..1....1..1..1....1..2..2....1..1..1....1..0..1
..2..1..2....2..1..2....2..1..3....2..1..2....3..2..2....2..1..2....2..2..2
..1..2..1....3..2..3....0..0..1....0..2..0....1..3..0....3..2..0....1..2..3

R. H. Hardin, Table of n, a(n) for n = 1..83

Main diagonal is A208346.

Columns 1..7 are A007581(n-1), A208347, A208348, A208349, A208350, A208351, A208352.

A222679 T(n,k)=Number of nXk 0..3 arrays with no more than floor(nXk/2) elements unequal 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, 1, 1, 5, 4, 5, 5, 15, 5, 10, 11, 18, 51, 14, 25, 58, 130, 63, 187, 17, 150, 142, 527, 670, 234, 715, 75, 359, 2398, 5374, 8276, 9587, 1163, 2795, 95, 2249, 6463, 58167, 69973, 133028, 51055, 4953, 11051, 411, 6511, 146947, 687222, 1999404, 3759625

Offset: 1

R. H. Hardin Feb 28 2013

Table starts
......1......1........1.........4..........5.........14.........17.........75
......2......1........5........10.........25........150........359.......2249
......5......5.......11........58........142.......2398.......6463.....146947
.....15.....18......130.......527.......5374......58167.....687222...11377192
.....51.....63......670......8276......69973....1999404...20997839.1097897602
....187....234.....9587....133028....3759625...96510910.3254033272
....715...1163....51055...2363628...65204276.5825299399
...2795...4953...832433..40206464.4282689196
..11051..24021..4403793.727306856
..43947.109395.76270010
.175275.536457
.700075

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

R. H. Hardin, Table of n, a(n) for n = 1..82

Column 1 is A007581(n-1)
Column 2 is A222373
Row 1 is A222372

A099262 a(n) = (1/5040)7^n + (1/240)5^n + (1/72)4^n + (1/16)3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).

Original entry on oeis.org

1, 2, 5, 15, 52, 203, 877, 4139, 21110, 115179, 665479, 4030523, 25343488, 164029595, 1084948961, 7291973067, 49582466986, 339971207051, 2345048898523, 16244652278171, 112871151708404, 785938550025147, 5480960778389365, 38264428799608235, 267342497477336542, 1868866831126685483

Offset: 1

Nelma Moreira, Oct 10 2004

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

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, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
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.
Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

Cf. A007051, A007581, A056272, A056273, A099263.

A row of the array in A278984.

Table[Sum[StirlingS2[n, k], {k, 0, 7}], {n, 1, 30}] (* Robert A. Russell, Apr 25 2018 *)

a(n) = (1/5040)*7^n + (1/240)*5^n + (1/72)*4^n + (1/16)*3^n + (11/60)*2^n + 53/144; \\ Altug Alkan, Apr 25 2018

For c=7, 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)!, c > 1, g(k, c) = g(k-1, c-1)/k, for c > 1 and 2 <= k <= c.
G.f.: -x*(531*x^5-881*x^4+535*x^3-151*x^2+20*x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Dec 05 2012
a(n) = Sum_{k=0..7} Stirling2(n,k).
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=7. - Robert A. Russell, Apr 25 2018

More terms from Michel Marcus, Jan 05 2025

cp's OEIS Frontend

A164864 Number of ways of placing n labeled balls into 10 indistinguishable boxes; word structures of length n using a 10-ary alphabet.

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A124303 Number of set partitions of length <= 4; sum of first 4 columns of triangle of Stirling numbers of 2nd kind; dimension of space of symmetric polynomials in 4 noncommuting variables.

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A241328 T(n,k)=Number of nXk 0..3 arrays with no element equal to fewer vertical neighbors than horizontal neighbors, with new values 0..3 introduced in row major order.

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A222878 T(n,k)=Number of nXk 0..3 arrays with no more than floor(nXk/2) elements equal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..3 order.

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A028401 The (2^n+1)-th triangular number (cf. A000217).

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A200801 T(n,k) is the number of n X k 0..3 arrays with values 0..3 introduced in row major order and each element equal to no more than two horizontal and vertical neighbors.

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A203647 T(n,k) = number of arrays of n 0..k integers with new values introduced in order 0..k but otherwise unconstrained. Array read by antidiagonals.

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A208353 T(n,k) is the number of n X k 0..3 arrays with new values 0..3 introduced in row major order and no element equal to any knight-move neighbor (colorings ignoring permutations of colors).

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A222679 T(n,k)=Number of nXk 0..3 arrays with no more than floor(nXk/2) elements unequal to at least one horizontal, diagonal or antidiagonal neighbor, with new values introduced in row major 0..3 order.

A099262 a(n) = (1/5040)7^n + (1/240)5^n + (1/72)4^n + (1/16)3^n + (11/60)*2^n + 53/144. Partial sum of Stirling numbers of second kind S(n,i), i=1..7 (i.e., a(n) = Sum_{i=1..7} S(n,i)).