A020557 -id:A020557 - cp's OEIS frontend

A208466 T(n,k) is the number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

1, 2, 2, 5, 15, 5, 15, 198, 203, 15, 52, 4041, 20746, 4140, 52, 203, 113458, 4132120, 4150760, 115975, 203, 877, 4132120, 1358524513, 10318694804, 1366230232, 4213597, 877, 4140, 187612143, 671329819215, 50996571454200, 51074630353994

Offset: 1

R. H. Hardin, Feb 27 2012

			Table starts:
....1...........2..................5.....................15
....2..........15................198...................4041
....5.........203..............20746................4132120
...15........4140............4150760............10318694804
...52......115975.........1366230232.........51074630353994
..203.....4213597.......675203938944.....441285917587055633
..877...190899322....470798015742024.6105599904286957450405
.4140.10480142147.442649055938121520
Some solutions for n=4 and k=3:
..0..1..2....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0....0..0..0
..0..0..1....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0
..0..0..0....0..0..2....2..0..0....2..0..1....1..1..0....0..2..0....0..0..0
..0..1..0....0..0..0....0..2..0....0..0..0....0..0..0....0..0..0....0..2..1

R. H. Hardin, Table of n, a(n) for n = 1..58

Column 1 and row 1 are A000110.

Column 2 is A020557.

A216460 T(n,k)=Number of horizontal, diagonal and antidiagonal neighbor colorings of the even squares of an nXk array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 7, 5, 5, 2, 5, 15, 20, 15, 5, 5, 15, 203, 203, 322, 52, 15, 5, 52, 716, 3429, 4140, 1335, 203, 15, 15, 203, 17733, 83440, 580479, 115975, 36401, 877, 52, 15, 877, 83440, 2711768, 18171918, 20880505, 4213597, 192713, 4140, 52, 52

Offset: 1

R. H. Hardin Sep 07 2012

Table starts
...1......1..........1............1...............2................2
...1......1..........1............2...............5...............15
...2......2..........7...........15.............203..............716
...2......5.........20..........203............3429............83440
...5.....15........322.........4140..........580479.........18171918
...5.....52.......1335.......115975........20880505.......6423127757
..15....203......36401......4213597......8195008751....3376465219485
..15....877.....192713....190899322....484968748793.2486327138729353
..52...4140....7712455..10480142147.348950573407587
..52..21147...49055292.682076806159
.203.115975.2659544320
.203.678570

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

R. H. Hardin, Table of n, a(n) for n = 1..96

Column 2 is A000110(n-1)
Column 4 is A020557(n-1)
Column 6 is A208051
Row 2 is A000110(n-2)
Odd squares: A216612

A216612 T(n,k)=Number of horizontal, diagonal and antidiagonal neighbor colorings of the odd squares of an nXk array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 5, 2, 2, 5, 15, 20, 15, 5, 2, 15, 41, 203, 67, 52, 5, 5, 52, 716, 3429, 4140, 1335, 203, 15, 5, 203, 2847, 83440, 83437, 115975, 6097, 877, 15, 15, 877, 83440, 2711768, 18171918, 20880505, 4213597, 192713, 4140, 52, 15, 4140

Offset: 1

R. H. Hardin Sep 10 2012

Table starts
...1......1.........1............1..............1................2
...1......1.........1............2..............5...............15
...1......2.........2...........15.............41..............716
...2......5........20..........203...........3429............83440
...2.....15........67.........4140..........83437.........18171918
...5.....52......1335.......115975.......20880505.......6423127757
...5....203......6097......4213597......942420901....3376465219485
..15....877....192713....190899322...484968748793.2486327138729353
..15...4140...1094076..10480142147.33862631596393
..52..21147..49055292.682076806159
..52.115975.329588907
.203.678570

			Some solutions for n=4 k=4
..x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1....x..0..x..1
..2..x..3..x....2..x..3..x....1..x..2..x....2..x..3..x....1..x..2..x
..x..4..x..2....x..1..x..2....x..0..x..3....x..4..x..0....x..3..x..4
..5..x..6..x....4..x..3..x....2..x..1..x....2..x..5..x....0..x..2..x

R. H. Hardin, Table of n, a(n) for n = 1..96

Column 2 is A000110(n-1)
Column 4 is A020557(n-1)
Column 6 is A208051
Row 2 is A000110(n-2)
Row 4 is A216462
Row 6 is A216464
Even squares: A216460

A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(nx) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

Original entry on oeis.org

1, 2, 1, 15, 12, 2, 203, 206, 60, 5, 4140, 4949, 1947, 298, 15, 115975, 156972, 75595, 16160, 1535, 52, 4213597, 6301550, 3528368, 945360, 127915, 8307, 203, 190899322, 310279615, 195764198, 62079052, 10690645, 1001567, 47397, 877, 10480142147, 18293310174, 12735957930, 4614975428, 952279230, 114741060, 7901236, 285096, 4140, 682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147, 51724158235372, 101557600812015, 82635818516305, 36672690416280, 9831937482310, 1665456655065, 180791918475, 12443391060, 520878315, 12004575, 115975

Offset: 0

Paul D. Hanna, Jul 20 2019

			E.g.f.: A(x,y) = 1 + (2 + y)*x + (15 + 12*y + 2*y^2)*x^2/2! + (203 + 206*y + 60*y^2 + 5*y^3)*x^3/3! + (4140 + 4949*y + 1947*y^2 + 298*y^3 + 15*y^4)*x^4/4! + (115975 + 156972*y + 75595*y^2 + 16160*y^3 + 1535*y^4 + 52*y^5)*x^5/5! + (4213597 + 6301550*y + 3528368*y^2 + 945360*y^3 + 127915*y^4 + 8307*y^5 + 203*y^6)*x^6/6! + (190899322 + 310279615*y + 195764198*y^2 + 62079052*y^3 + 10690645*y^4 + 1001567*y^5 + 47397*y^6 + 877*y^7)*x^7/7! + (10480142147 + 18293310174*y + 12735957930*y^2 + 4614975428*y^3 + 952279230*y^4 + 114741060*y^5 + 7901236*y^6 + 285096*y^7 + 4140*y^8)*x^8/8! + (682076806159 + 1267153412532*y + 959061013824*y^2 + 387848415927*y^3 + 92381300277*y^4 + 13455280629*y^5 + 1200540180*y^6 + 63424134*y^7 + 1805067*y^8 + 21147*y^9)*x^9/9! + (51724158235372 + 101557600812015*y + 82635818516305*y^2 + 36672690416280*y^3 + 9831937482310*y^4 + 1665456655065*y^5 + 180791918475*y^6 + 12443391060*y^7 + 520878315*y^8 + 12004575*y^9 + 115975*y^10)*x^10/10! + ...
such that
A(x,y) = exp(-1-y) * (1 + (exp(x) + y) + (exp(2*x) + y)^2/2! + (exp(3*x) + y)^3/3! + (exp(4*x) + y)^4/4! + (exp(5*x) + y)^5/5! + (exp(6*x) + y)^6/6! + ...)
also
A(x,y) = exp(-1-y) * (exp(y) + exp(x)*exp(y*exp(x)) + exp(4*x)*exp(y*exp(2*x))/2! + exp(9*x)*exp(y*exp(3*x))/3! + exp(16*x)*exp(y*exp(4*x))/4! + exp(25*x)*exp(y*exp(5*x))/5! + exp(36*x)*exp(y*exp(6*x))/6! + ...).
This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(x,y) begins:
[1],
[2, 1],
[15, 12, 2],
[203, 206, 60, 5],
[4140, 4949, 1947, 298, 15],
[115975, 156972, 75595, 16160, 1535, 52],
[4213597, 6301550, 3528368, 945360, 127915, 8307, 203],
[190899322, 310279615, 195764198, 62079052, 10690645, 1001567, 47397, 877],
[10480142147, 18293310174, 12735957930, 4614975428, 952279230, 114741060, 7901236, 285096, 4140],
[682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147], ...
Main diagonal is A000110 (Bell numbers).
Leftmost column is A020557(n) = A000110(2*n), for n >= 0.
Row sums form A326433.

Paul D. Hanna, Table of n, a(n) for n = 0..495 (first 30 rows of this triangle).

Cf. A000110, A020557, A108459, A326433, A326434, A326435, A326436, A326437.

Cf. A326601 (central terms).

E.g.f.: exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!.
E.g.f.: exp(-1-y) * Sum_{n>=0} exp(n^2*x) * exp( y*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
T(n,n) = A000110(n) for n >= 0, where A000110 is the Bell numbers.
T(n,0) = A000110(2*n) for n >= 0, where A000110 is the Bell numbers.
Sum_{k=0..n} T(n,k) * (-1)^k = A108459(n) for n >= 0.
Sum_{k=0..n} T(n,k) = A326433(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 2^k = A326434(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 3^k = A326435(n) for n >= 0.
Sum_{k=0..n} T(n,k) * 4^k = A326436(n) for n >= 0.

A208054 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 15, 15, 5, 15, 203, 716, 203, 15, 52, 4140, 83440, 83440, 4140, 52, 203, 115975, 18171918, 112073062, 18171918, 115975, 203, 877, 4213597, 6423127757, 346212384169, 346212384169, 6423127757, 4213597, 877, 4140, 190899322

Offset: 1

R. H. Hardin, Feb 22 2012

Equivalently, the number of colorings in the rhombic hexagonal square grid graph RH_(n,k) using any number of colors up to permutation of the colors. - Andrew Howroyd, Jun 25 2017

			Table starts
...1.........1.............2................5................15
...1.........2............15..............203..............4140
...2........15...........716............83440..........18171918
...5.......203.........83440........112073062......346212384169
..15......4140......18171918.....346212384169.18633407199331522
..52....115975....6423127757.2043836452962923
.203...4213597.3376465219485
.877.190899322
...
Some solutions for n=4 k=3
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0
..2..3..1....2..3..4....2..3..2....2..3..1....2..3..0....2..3..1....2..3..2
..4..2..4....0..5..0....0..4..0....0..4..5....4..5..3....4..5..3....0..1..4
..0..5..0....1..2..1....1..2..1....5..3..4....0..1..0....0..6..4....2..0..1

Andrew Howroyd, Table of n, a(n) for n = 1..231 (terms 1..49 from R. H. Hardin)

Columns 1-5 are A000110(n-1), A020557(n-1), A208051, A208052, A208053.

Cf. A207868, A212162, A212163, A208050, A068271.

A326433 E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.

Original entry on oeis.org

1, 3, 29, 474, 11349, 366289, 15125300, 770762673, 47199596441, 3403242019876, 284281430425747, 27150503912943937, 2932403885598294838, 354869660881411722107, 47739034071736749352125, 7090201955561116768761250, 1155624866838027573814278801, 205611555585528308269669174557, 39746979329229607204823274477284

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 1, r = 1.

			E.g.f.: A(x) = 1 + 3*x + 29*x^2/2! + 474*x^3/3! + 11349*x^4/4! + 366289*x^5/5! + 15125300*x^6/6! + 770762673*x^7/7! + 47199596441*x^8/8! + 3403242019876*x^9/9! + 284281430425747*x^10/10! + 27150503912943937*x^11/11! + 2932403885598294838*x^12/12! + ...
such that
A(x) = exp(-2) * (1 + (exp(x) + 1) + (exp(2*x) + 1)^2/2! + (exp(3*x) + 1)^3/3! + (exp(4*x) + 1)^4/4! + (exp(5*x) + 1)^5/5! + (exp(6*x) + 1)^6/6! + ...)
also
A(x) = exp(-2) * (exp(1) + exp(x)*exp(exp(x)) + exp(4*x)*exp(exp(2*x))/2! + exp(9*x)*exp(exp(3*x))/3! + exp(16*x)*exp(exp(4*x))/4! + exp(25*x)*exp(exp(5*x))/5! + exp(36*x)*exp(exp(6*x))/6! + ...).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A326600, A020557, A326430, A326434, A326435, A326436, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-2) * sum(n=0,500, (exp(n*x +O(x^31)) + 1)^n/n! ) )))

E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.
E.g.f.: exp(-2) * Sum_{n>=0} exp(n^2*x) * exp( exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n) = 0 (mod 2), a(3*n-1) = 1 (mod 2), and a(3*n-2) = 1 (mod 2) for n > 0.

A282010 Number of ways to partition Turan graph T(2n,n) into connected components.

Original entry on oeis.org

1, 1, 12, 163, 3411, 97164, 3576001, 163701521, 9064712524, 594288068019, 45352945127123, 3973596101084108, 395147058261233761, 44170986458602383553, 5504694207040057913164, 759355292729159336345955, 115228949414563130433140659, 19129024114529146183236435660

Offset: 0

Tengiz Gogoberidze, Feb 04 2017

Turan graph T(2n,n) is also called cocktail party graph, so a(n) is the number of ways to seat n married couples for one or a few tables in such a manner that no table is fully occupied by any couple.

If we dissect (n-1)-skeleton of n-cube along some (n-2)-edges into some parts, then a(n) is the number of ways of such dissections.

			For n=1, Turan graph T(2,1) (2-empty graph) shall be partitioned into two singleton subgraphs (1 way), a(1)=1.
For n=2, Turan graph T(4,2) (square graph) shall be partitioned into: the same square graph (1 way) or one singleton + one 3-path subgraphs (4 ways) or two singleton + one 2-path subgraphs (4 ways) or two 2-path subgraphs (2 ways) or four singleton subgraphs (1 way), a(2)=12.

Indranil Ghosh, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Turan Graph

Cf. A000110, A020557

A282010 := proc(n)
    add((-1)^(n-j)*combinat[bell](2*j)*binomial(n,j),j=0..n) ;
end proc:
seq(A282010(n),n=0..20) ; # R. J. Mathar, Jun 27 2024

a[n_]:=BellB[2n];Table[Sum[((-1)^(n-j))*a[j]*Binomial[n,j],{j,0,n}],{n,0,17}] (* Indranil Ghosh, Feb 25 2017 *)

bell(n) = polcoeff( sum( k=0, n, prod(i=1, k, x/(1 - i*x)), x^n * O(x)), n)
a(n) = sum(j=0, n, ((-1)^(n-j))*bell(2*j)*binomial(n,j)); \\ Michel Marcus, Feb 05 2017

a(n) = Sum_{j=0..n} ((-1)^(n-j))*A020557(j)*binomial(n,j).

a(n) = Sum_{j=0..n} ((-1)^(n-j))*A000110(2*j)*binomial(n,j).

More terms from Michel Marcus, Feb 05 2017

A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.

Original entry on oeis.org

1, 4, 47, 895, 24450, 887803, 40818505, 2297393888, 154381810471, 12149510583583, 1102672816721422, 113974516318639363, 13277046519634998953, 1727765194711759098324, 249264545884060054668295, 39606622952407779396832791, 6891271396238954765341535650, 1306288225868329080524305347859, 268542657134280438710389415260401, 59628381166607045580114829853101712

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 2, r = 1.

			E.g.f.: A(x) = 1 + 4*x + 47*x^2/2! + 895*x^3/3! + 24450*x^4/4! + 887803*x^5/5! + 40818505*x^6/6! + 2297393888*x^7/7! + 154381810471*x^8/8! + 12149510583583*x^9/9! + 1102672816721422*x^10/10! + ...
such that
A(x) = exp(-3) * (1 + (exp(x) + 2) + (exp(2*x) + 2)^2/2! + (exp(3*x) + 2)^3/3! + (exp(4*x) + 2)^4/4! + (exp(5*x) + 2)^5/5! + (exp(6*x) + 2)^6/6! + ...)
also
A(x) = exp(-3) * (exp(2) + exp(x)*exp(2*exp(x)) + exp(4*x)*exp(2*exp(2*x))/2! + exp(9*x)*exp(2*exp(3*x))/3! + exp(16*x)*exp(2*exp(4*x))/4! + exp(25*x)*exp(2*exp(5*x))/5! + exp(36*x)*exp(2*exp(6*x))/6! + ...).

Cf. A326600, A020557, A326433, A326435, A326436, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-3) * sum(n=0, 500, (exp(n*x +O(x^31)) + 2)^n/n! ) )))

E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.
E.g.f.: exp(-3) * Sum_{n>=0} exp(n^2*x) * exp( 2*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.

A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.

Original entry on oeis.org

1, 5, 69, 1496, 45771, 1840537, 92925982, 5705543791, 416015394341, 35365673566750, 3454046493504337, 382930667897753421, 47708365129614794580, 6622948820406278058625, 1016977626656613380728781, 171637260767262574245781800, 31661205827344145981298200207, 6352045190999137085697971335893

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 3, r = 1.

			E.g.f.: A(x) = 1 + 5*x + 69*x^2/2! + 1496*x^3/3! + 45771*x^4/4! + 1840537*x^5/5! + 92925982*x^6/6! + 5705543791*x^7/7! + 416015394341*x^8/8! + 35365673566750*x^9/9! + 3454046493504337*x^10/10! + ...
such that
A(x) = exp(-4) * (1 + (exp(x) + 3) + (exp(2*x) + 3)^2/2! + (exp(3*x) + 3)^3/3! + (exp(4*x) + 3)^4/4! + (exp(5*x) + 3)^5/5! + (exp(6*x) + 3)^6/6! + ...)
also
A(x) = exp(-4) * (exp(3) + exp(x)*exp(3*exp(x)) + exp(4*x)*exp(3*exp(2*x))/2! + exp(9*x)*exp(3*exp(3*x))/3! + exp(16*x)*exp(3*exp(4*x))/4! + exp(25*x)*exp(3*exp(5*x))/5! + exp(36*x)*exp(3*exp(6*x))/6! + ...).

Cf. A326600, A020557, A326433, A326434, A326436, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-4) * sum(n=0, 500, (exp(n*x +O(x^31)) + 3)^n/n! ) )))

E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.
E.g.f.: exp(-4) * Sum_{n>=0} exp(n^2*x) * exp( 3*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n) = 0 (mod 2), a(3*n-1) = 1 (mod 2), and a(3*n-2) = 1 (mod 2) for n > 0.

A326436 E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.

Original entry on oeis.org

1, 6, 95, 2307, 78000, 3433831, 188460821, 12508220886, 981371259995, 89426179550623, 9331384489007032, 1102143627943740931, 145924317814992561097, 21480095845779426077750, 3490477008130417972086807, 622292123277813938275834747, 121062971468108753273621477712, 25577093024015935514169919403295

Offset: 0

Paul D. Hanna, Jul 11 2019

nonn

More generally, the following sums are equal:
(1) exp(-(p+1)*r) * Sum_{n>=0} (q^n + p)^n * r^n / n!,
(2) exp(-(p+1)*r) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,
here, q = exp(x), p = 4, r = 1.

			E.g.f.: A(x) = 1 + 6*x + 95*x^2/2! + 2307*x^3/3! + 78000*x^4/4! + 3433831*x^5/5! + 188460821*x^6/6! + 12508220886*x^7/7! + 981371259995*x^8/8! + 89426179550623*x^9/9! + 9331384489007032*x^10/10! + ...
such that
A(x) = exp(-5) * (1 + (exp(x) + 4) + (exp(2*x) + 4)^2/2! + (exp(3*x) + 4)^3/3! + (exp(4*x) + 4)^4/4! + (exp(5*x) + 4)^5/5! + (exp(6*x) + 4)^6/6! + ...)
also
A(x) = exp(-5) * (exp(4) + exp(x)*exp(4*exp(x)) + exp(4*x)*exp(4*exp(2*x))/2! + exp(9*x)*exp(4*exp(3*x))/3! + exp(16*x)*exp(4*exp(4*x))/4! + exp(25*x)*exp(4*exp(5*x))/5! + exp(36*x)*exp(4*exp(6*x))/6! + ...).

Cf. A326600, A020557, A326433, A326434, A326435, A326437.

/* Requires suitable precision */
\p200
Vec(round(serlaplace( exp(-5) * sum(n=0, 500, (exp(n*x +O(x^31)) + 4)^n/n! ) )))

E.g.f.: exp(-5) * Sum_{n>=0} (exp(n*x) + 4)^n / n!.
E.g.f.: exp(-5) * Sum_{n>=0} exp(n^2*x) * exp( 4*exp(n*x) ) / n!.
FORMULAS FOR TERMS.
a(3*n+1) = 0 (mod 2), a(3*n) = 1 (mod 2), and a(3*n+2) = 1 (mod 2) for n >= 0.

cp's OEIS Frontend

A208466 T(n,k) is the number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to more than two of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A216460 T(n,k)=Number of horizontal, diagonal and antidiagonal neighbor colorings of the even squares of an nXk array with new integer colors introduced in row major order.

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A216612 T(n,k)=Number of horizontal, diagonal and antidiagonal neighbor colorings of the odd squares of an nXk array with new integer colors introduced in row major order.

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A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

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A208054 T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

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A326433 E.g.f.: exp(-2) * Sum_{n>=0} (exp(n*x) + 1)^n / n!.

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A282010 Number of ways to partition Turan graph T(2n,n) into connected components.

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Maple

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A326434 E.g.f.: exp(-3) * Sum_{n>=0} (exp(n*x) + 2)^n / n!.

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A326435 E.g.f.: exp(-4) * Sum_{n>=0} (exp(n*x) + 3)^n / n!.

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A326600 E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(nx) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.