A109808 -id:A109808 - cp's OEIS frontend

A321982 Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

2, 0, 12, 2, 0, 0, 0, 54, 26, 16, 0, 2, 0, 0, 0, 0, 0, 0, 216, 120, 168, 84, 0, 24, 40, 32, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 810, 648, 822, 56, 240, 870, 280, 282, 120, 24, 0, 266, 232, 0, 48, 0, 54, 0, 48, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).
The n-ladder has 2*n vertices and looks like:
  o-o-o-   -o
  | | | ... |
  o-o-o-   -o
Conjecture: All terms are nonnegative (verified up to the 5-ladder).

			Triangle begins:
    2   0
   12   2   0   0   0
   54  26  16   0   2   0   0   0   0   0   0
  216 120 168  84   0  24  40  32   0   0   2   0   0   [+9 more zeros]
For example, row 3 gives: X_L3 = 54e(6) + 26e(42) + 16e(51) + 2e(222).

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Row sums are A109808.

Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321980, A321981.

A207864 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 4, 34, 500, 10900, 322768, 12297768, 580849872, 33093252880, 2227152575552, 174131286983712, 15604440074084672, 1584856558077903168, 180712593036822482176, 22946861101272125055616, 3222156375409363475703040

Offset: 1

R. H. Hardin, Feb 21 2012

nonn

From Gus Wiseman, Mar 01 2019: (Start)
Also the number of stable partitions of the n-ladder graph. A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The n-ladder has 2n vertices and looks like:
  o-o-o-   -o
  | | | ... |
  o-o-o-   -o
(End)

			Some solutions for n=5:
  0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1
  1 0   1 0   1 2   1 2   1 0   1 0   1 2   1 0   1 0   1 0
  0 1   0 1   0 1   0 1   2 1   0 1   0 1   0 2   2 1   0 1
  1 2   1 0   1 0   1 3   3 0   2 0   3 2   2 1   1 0   1 2
  0 1   0 1   2 1   2 4   1 2   0 1   0 1   0 2   0 1   2 0

R. H. Hardin, Table of n, a(n) for n = 1..61

Column 2 of A207868.

Cf. A000110, A000569, A109808, A229048, A240936, A321750, A321979, A321980, A321982, A322064.

Table[Expand[x*(x-1)*(x^2-3*x+3)^(n-1)]/.x^k_.->BellB[k],{n,20}] (* Gus Wiseman, Mar 01 2019 *)

It appears that the sequence terms are given by the Dobinski-type formula a(n+1) = (1/e) * Sum_{k>=0} (1+k+k^2)^n/k!. - Peter Bala, Mar 12 2012

Apply x^n -> B(n) to the polynomial chi(n) = x (x - 1) (x^2 - 3 x + 3)^(n - 1), where B = A000110. - Gus Wiseman, Mar 01 2019

A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

Original entry on oeis.org

4, 5, 8, 7, 6, 6, 7, 2, 6, 6, 9, 9, 7, 6, 8, 9, 8, 5, 0, 2, 0, 0, 0, 5, 1, 5, 3, 3, 6, 9, 7, 4, 3, 7, 2, 1, 7, 8, 2, 5, 4, 6, 6, 8, 8, 7, 1, 4, 7, 3, 1, 8, 7, 0, 0, 7, 8, 2, 4, 4, 0, 1, 3, 8, 5, 0, 6, 9, 9, 7, 4, 4, 0, 3, 2, 6, 5, 9, 3, 0, 3, 6, 5, 2, 3, 7, 8, 1, 7, 1, 0, 9, 0, 4, 0, 5, 8, 4, 7, 5, 9, 8, 2

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.4587667266997689850200...

Cf. A048651, A098844, A067080, A132019, A132026, A132031, A132035, A000217, A109808.

digits = 103; NProduct[1-1/(2*7^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/7], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_7(n))} floor(n/7^k)*7^k/n.
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^(1/2*(1+floor(log_7(n)))*floor(log_7(n))).
Equals lim inf_{n->oo} A132031(n)/n^(1+floor(log_7(n)))*7^A000217(floor(log_7(n))).
Equals 1/2*exp(-Sum_{n>0} 7^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132031(n)/A132031(n+1).
Equals Product_{n>=1} (1 - 1/A109808(n)). - Amiram Eldar, May 08 2023

A324266 a(n) = 2*49^n.

Original entry on oeis.org

2, 98, 4802, 235298, 11529602, 564950498, 27682574402, 1356446145698, 66465861139202, 3256827195820898, 159584532595224002, 7819642097165976098, 383162462761132828802, 18774960675295508611298, 919973073089479921953602, 45078680581384516175726498, 2208855348487841292610598402

Offset: 0

Stefano Spezia, Feb 20 2019

x = A324265(n) and y = a(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 7^(6*n+1) = 4*y^3 (see Theorem 2.1 in Chakraborty, Hoque and Sharma).

			For A324265(0) = 5 and a(0) = 2, 5^2 + 7 = 32 = 4*2^3.

K. Chakraborty, A. Hoque, R. Sharma, Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, arXiv:1812.11874 [math.NT], 2018.

Cf. A324265 (5*343^n), A000290 (n^2), A000578 (n^3), A109808 (2*7^(n-1)).

GAP
```
List([0..20], n->2*49^n);
```
Magma
```
[2*49^n: n in [0..20]];
```
Maple
```
a:=n->2*49^n: seq(a(n), n=0..20);
```
Mathematica
```
2*49^Range[0,20]
```
PARI
```
a(n) = 2*49^n;
```

O.g.f.: 2/(1 - 49*x).
E.g.f.: 2*exp(49*x).
a(n) = 49*a(n-1) for n > 0.
a(n) = (49/2)*(A109808(n))^2.

A245807 a(n) = 7^n + 10^n.

Original entry on oeis.org

2, 17, 149, 1343, 12401, 116807, 1117649, 10823543, 105764801, 1040353607, 10282475249, 101977326743, 1013841287201, 10096889010407, 100678223072849, 1004747561509943, 10033232930569601, 100232630513987207, 1001628413597910449, 10011398895185373143

Offset: 0

Vincenzo Librandi, Aug 04 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-70).

Cf. 7^n+k^n: A034491 (k=1), A074602 (k=2), A074608 (k=3), A074613 (k=4), A074616 (k=5), A074619 (k=6), A109808 (k=7), A074622 (k=8), A074623 (k=9), this sequence (k=10).

Magma
```
[7^n+10^n: n in [0..25]];
```

I:=[2,17]; [n le 2 select I[n] else 17*Self(n-1)-70*Self(n-2): n in [1..25]];

Table[(7^n + 10^n), {n, 0, 30}] (* or *) CoefficientList[Series[(2 - 17 x)/((1 - 7 x) (1 - 10 x)), {x, 0, 40}], x]

G.f.: (2-17*x)/((1-7*x)*(1-10*x)).
E.g.f.: e^(7*x) + e^(10*x).
a(n) = 17*a(n-1)-70*a(n-2).
a(n) = A000420(n) + A011557(n).

A360922 Array read by antidiagonals: T(m,n) is the number of acyclic orientations in the grid graph P_m X P_n.

Original entry on oeis.org

1, 2, 2, 4, 14, 4, 8, 98, 98, 8, 16, 686, 2398, 686, 16, 32, 4802, 58670, 58670, 4802, 32, 64, 33614, 1435414, 5015972, 1435414, 33614, 64, 128, 235298, 35118638, 428816558, 428816558, 35118638, 235298, 128, 256, 1647086, 859207558, 36659327366, 128091434266, 36659327366, 859207558, 1647086, 256

Offset: 1

Andrew Howroyd, Mar 07 2023

			Array begins:
=====================================================
m\n|  1     2        3           4              5 ...
---+-------------------------------------------------
1  |  1     2        4           8             16 ...
2  |  2    14       98         686           4802 ...
3  |  4    98     2398       58670        1435414 ...
4  |  8   686    58670     5015972      428816558 ...
5  | 16  4802  1435414   428816558   128091434266 ...
6  | 32 33614 35118638 36659327366 38261306901842 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..253 (first 22 antidiagonals).
Eric Weisstein's World of Mathematics, Grid Graph.
Wikipedia, Acyclic orientation.

Main diagonal is A080690.
Rows 1..2 are A000079(n-1), A109808.
Cf. A116469 (spanning trees), A178435, A207868 (unlabeled colorings).

T(m,n) = T(n,m).

A324267 a(n) = 117^(5n).

Original entry on oeis.org

11, 184877, 3107227739, 52223176609373, 877714929273732011, 14751754816303613908877, 247932743197614838966495739, 4167005614922312598509893885373, 70034863369999307843155786531464011, 1177075948659578366919919304234315632877, 19783115469121533612823083746266142841763739

Offset: 0

Stefano Spezia, Feb 26 2019

x = a(n) and y = A324266(n) satisfy the Lebesgue-Ramanujan-Nagell equation x^2 + 7^(10*n+1) = 4*y^5 (see Theorem 2.1 in Chakraborty, Hoque and Sharma).

			For a(0) = 11 and A324266(0) = 2, 11^2 + 7 = 128 = 4*2^5.

K. Chakraborty, A. Hoque, R. Sharma, Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, arXiv:1812.11874 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (16807).

Cf. A324266: 2*49^n; A000290: n^2; A000584: n^5; A109808: 2*7^(n-1).

GAP
```
List([0..20], n->11*16807*n);
```
Magma
```
[11*16807^n: n in [0..20]];
```
Maple
```
a:=n->11*16807^n: seq(a(n), n=0..20);
```
Mathematica
```
11*16807^Range[0,20]
```
PARI
```
a(n) = 11*16807^n;
```

a(n) = 11*16807^n.
O.g.f.: 11/(1 - 16807*x).
E.g.f.: 11*exp(16807*x).
a(n) = 16807*a(n-1) for n > 0.
a(n) = 11*((7/2)*A109808(n))^5.

cp's OEIS Frontend

A321982 Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

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A207864 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A132023 Decimal expansion of Product_{k>=0} 1-1/(2*7^k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A324266 a(n) = 2*49^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A245807 a(n) = 7^n + 10^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

A360922 Array read by antidiagonals: T(m,n) is the number of acyclic orientations in the grid graph P_m X P_n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A324267 a(n) = 11*7^(5*n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

A324267 a(n) = 117^(5n).