A016161 -id:A016161 - cp's OEIS frontend

A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

0, 1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509, 39848449432985688, 279034513462540441, 1953718431395986212

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A081199.
Conjecture (verified up to a(9)): Number of collinear 5-tuples of points in a 5 X 5 X 5 X ... n-dimensional cubic grid. - Ron Hardin, May 24 2010
a(n) is also the total number of words of length n, over an alphabet of seven letters, of which one of them appears an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5-, 6- and 8-letter case analogs see A131577, A003462, A005059, A081199, A081201 respectively. - Wolfdieter Lang, Jul 17 2017

			The a(2) = 12 words of length 2 over {A, B, C, D, E, F, G} with say, A, appearing an odd number of times (that is once) are: AB, AC, AD, AE, AF, AG; BA, CA, DA, EA, FA, GA. - _Wolfdieter Lang_, Jul 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Index entries for linear recurrences with constant coefficients, signature (12,-35).

Cf. A000035, A003462, A005059, A006516, A081199, A081201 (binomial transform, and 8-letter analog), A121213, A131577.

Apart from offset same as A016161.

[7^n/2-5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x / ((1 - 5 x) (1 - 7 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{12,-35},{0,1},30] (* Harvey P. Dale, Feb 07 2014 *)

[lucas_number1(n,12,35) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009

a(n) = 12*a(n-1) - 35*a(n-2), a(0) = 0, a(1) = 1.
G.f.: x/((1-5*x)*(1-7*x)).
a(n) = 7^n/2 - 5^n/2.
a(n) = Sum_{k=0..n-1} 7^k * 5^(n-k-1), with a(0)=0. - Reinhard Zumkeller, Aug 01 2010
a(n) = A121213(n)/2. - Reinhard Zumkeller, Aug 01 2010
E.g.f.: exp(5*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jun 19 2021

A016164 Expansion of 1/((1-5x)(1-10*x)).

Original entry on oeis.org

1, 15, 175, 1875, 19375, 196875, 1984375, 19921875, 199609375, 1998046875, 19990234375, 199951171875, 1999755859375, 19998779296875, 199993896484375, 1999969482421875, 19999847412109375, 199999237060546875

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..994
Index entries for linear recurrences with constant coefficients, signature (15,-50).

Second column of triangle A075500.

Cf. A008277, A016161, A075911.

[n le 2 select 15^(n-1) else 15*Self(n-1) -50*Self(n-2): n in [1..31]]; // G. C. Greubel, Nov 09 2024

Table[5^n*(2^(n+1)-1), {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{15,-50},{1,15},20] (* Harvey P. Dale, Aug 08 2023 *)

Vec(1/((1-5*x)(1-10*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

A016164=BinaryRecurrenceSequence(15,-50,1,15)
[A016164(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

a(n) = (5^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = -5^n + 2*10^n.
G.f.: 1/((1-5*x)*(1-10*x)).
E.g.f.: (d^2/dx^2)((((exp(5*x)-1)/5)^2)/2!) = -exp(5*x) + 2*exp(10*x).
Sum_{k=1..n} 5^(k-1)*5^(n-k)*binomial(n, k). - Zerinvary Lajos, Sep 24 2006
a(0)=1, a(n) = 10*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011

A016162 Expansion of 1/((1-5x)(1-8*x)).

Original entry on oeis.org

1, 13, 129, 1157, 9881, 82173, 673009, 5462197, 44088201, 354658733, 2847035489, 22825112037, 182845036921, 1463980998493, 11717951503569, 93774129606677, 750345624744041, 6003527937405453, 48032038196509249

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-40).

Cf. A016161.

[n le 2 select 13^(n-1) else 13*Self(n-1) -40*Self(n-2): n in [1..31]]; // G. C. Greubel, Nov 09 2024

Table[(8^(n+1)-5^(n+1))/3, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/((1-5x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {13,-40},{1,13},30] (* Harvey P. Dale, Feb 02 2015 *)

Vec(1/((1-5*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

A016162=BinaryRecurrenceSequence(13,-40,1,13)
[A016162(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

a(n) = (8^(n+1) - 5^(n+1))/3. - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005
a(0)=1, a(n) = 8*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011
a(0)=1, a(1)=13, a(n) = 13*a(n-1) - 40*a(n-2). - Harvey P. Dale, Feb 02 2015
E.g.f.: (1/3)*(-5*exp(5*x) + 8*exp(8*x)). - G. C. Greubel, Nov 09 2024

A162590 Polynomials with e.g.f. exp(x*t)/csch(t), triangle of coefficients read by rows.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 0, 4, 0, 4, 0, 1, 0, 10, 0, 5, 0, 0, 6, 0, 20, 0, 6, 0, 1, 0, 21, 0, 35, 0, 7, 0, 0, 8, 0, 56, 0, 56, 0, 8, 0, 1, 0, 36, 0, 126, 0, 84, 0, 9, 0, 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 0, 1, 0, 55, 0, 330, 0, 462, 0, 165, 0, 11, 0, 0, 12, 0, 220, 0, 792, 0, 792, 0

Offset: 0

Peter Luschny, Jul 07 2009

Comment from Peter Bala (Dec 06 2011): "Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of (I-t*M)^-1 (apart from the initial 1) lists the row polynomials for" A196776(n,k), which gives the number of ordered partitions of an n set into k odd-sized blocks. - Peter Luschny, Dec 06 2011

The n-th row of the triangle is formed by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probability of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms in the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k) as the (2k)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k). - Luca Onnis, Oct 29 2023

			Triangle begins:
  0
  1,  0
  0,  2,  0
  1,  0,  3,  0
  0,  4,  0,  4,  0
  1,  0, 10,  0,  5,  0
  0,  6,  0, 20,  0,  6,  0
  1,  0, 21,  0, 35,  0,  7,  0
  ...
  p[0](x) = 0;
  p[1](x) = 1
  p[2](x) = 2*x
  p[3](x) = 3*x^2 +  1
  p[4](x) = 4*x^3 +  4*x
  p[5](x) = 5*x^4 + 10*x^2 +  1
  p[6](x) = 6*x^5 + 20*x^3 +  6*x
  p[7](x) = 7*x^6 + 35*x^4 + 21*x^2 + 1
  p[8](x) = 8*x^7 + 56*x^5 + 56*x^3 + 8*x
.
Cf. the triangle of odd-numbered terms in rows of Pascal's triangle (A034867).
p[n] (k), n=0,1,...
k=0:  0, 1,  0,   1,    0,     1, ... A000035, (A059841)
k=1:  0, 1,  2,   4,    8,    16, ... A131577, (A000079)
k=2:  0, 1,  4,  13,   40,   121, ... A003462
k=3:  0, 1,  6,  28,  120,   496, ... A006516
k=4:  0, 1,  8,  49,  272,  1441, ... A005059
k=5:  0, 1, 10,  76,  520,  3376, ... A081199, (A016149)
k=6:  0, 1, 12, 109,  888,  6841, ... A081200, (A016161)
k=7:  0, 1, 14, 148, 1400, 12496, ... A081201, (A016170)
k=8:  0, 1, 16, 193, 2080, 21121, ... A081202, (A016178)
k=9:  0, 1, 18, 244, 2952, 33616, ... A081203, (A016186)
k=10: 0, 1, 20, 301, 4040, 51001, ... ......., (A016190)
.
p[n] (k), k=0,1,...
p[0]: 0,  0,   0,    0,    0,     0, ... A000004
p[1]: 1,  1,   1,    1,    1,     1, ... A000012
p[2]: 0,  2,   4,    6,    8,    10, ... A005843
p[3]: 1,  4,  13,   28,   49,    76, ... A056107
p[4]: 0,  8,  40,  120,  272,   520, ... A105374
p[5]: 1, 16, 121,  496, 1441,  3376, ...
p[6]: 0, 32, 364, 2016, 7448, 21280, ...

Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.
Luca Onnis, Animation of the Ehrenfest model.
Wikipedia, Ehrenfest model.

Cf. A119467.

# Polynomials: p_n(x)
p := proc(n,x) local k;
pow := (n,k) -> `if`(n=0 and k=0,1,n^k);
add((k mod 2)*binomial(n,k)*pow(x,n-k),k=0..n) end;
# Coefficients: a(n)
seq(print(seq(coeff(i!*coeff(series(exp(x*t)/csch(t), t,16),t,i),x,n), n=0..i)), i=0..8);

p[n_, x_] := Sum[Binomial[n, 2*k-1]*x^(n-2*k+1), {k, 0, n+2}]; row[n_] := CoefficientList[p[n, x], x] // Append[#, 0]&; Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
n = 15; "n-th row"
mat = Table[Table[0, {j, 1, n + 1}], {i, 1, n + 1}];
mat[[1, 2]] = 1;
mat[[n + 1, n]] = 1;
For[i = 2, i <= n, i++, mat[[i, i - 1]] = (i - 1)/n ];
For[i = 2, i <= n, i++, mat[[i, i + 1]] = (n - i + 1)/n];
mat // MatrixForm;
P2 = Dot[mat, mat];
R1 = Simplify[
  Eigenvectors[Transpose[P2]][[1]]/
   Total[Eigenvectors[Transpose[P2]][[1]]]]
R2 = Table[Dot[R1, Transpose[mat][[k]]], {k, 1, n + 1}]
even = R1*2^(n - 1) (* Luca Onnis, Oct 29 2023 *)

p_n(x) = Sum_{k=0..n} (k mod 2)*binomial(n,k)*x^(n-k).
E.g.f.: exp(x*t)/csch(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2+1)*(t^3/3!) + ...
The 'co'-polynomials with generating function exp(x*t)*sech(t) are the Swiss-Knife polynomials (A153641).

A016163 Expansion of 1/((1-5x)(1-9*x)).

Original entry on oeis.org

1, 14, 151, 1484, 13981, 128954, 1176211, 10664024, 96366841, 869254694, 7833057871, 70546348964, 635161281301, 5717672234834, 51465153629131, 463216900240304, 4169104690053361, 37522705149933374, 337708161046665991

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-45).

Cf. A016142, A016161.

[(9^(n+1) - 5^(n+1))/4: n in [0..30]]; // G. C. Greubel, Nov 09 2024

Table[(9^(n+1) - 5^(n+1))/4, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)

def A016163(n): return (9^(n+1) - 5^(n+1))/4
[A016163(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

a(n) = ((7+sqrt4)^n - (7-sqrt4)^n)/4. Offset 1. a(3)=151. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) = 14*a(n-1) - 45*a(n-2). - Philippe Deléham, Jan 01 2009
a(0)=1, a(n) = 9*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011
From G. C. Greubel, Nov 09 2024: (Start)
a(n) = (9^(n+1) - 5^(n+1))/4.
E.g.f.: (1/4)*(9*exp(9*x) - 5*exp(5*x)). (End)

A016165 Expansion of 1/((1-5x)(1-11*x)).

Original entry on oeis.org

1, 16, 201, 2336, 26321, 292656, 3234841, 35661376, 392665761, 4321276496, 47543807081, 523030706016, 5753581906801, 63290621677936, 696202941972921, 7658262879280256, 84241044259973441, 926652249799160976

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..959
Index entries for linear recurrences with constant coefficients, signature (16,-55).

Cf. A016161.

[n le 2 select 16^(n-1) else 16*Self(n-1) -55*Self(n-2): n in [1..31]]; // G. C. Greubel, Nov 10 2024

Table[(11^(n+1)-5^(n+1))/6, {n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
CoefficientList[Series[1/((1-5x)(1-11x)),{x,0,30}],x] (* or *) LinearRecurrence[{16,-55},{1,16},30] (* Harvey P. Dale, Nov 24 2021 *)

A016165=BinaryRecurrenceSequence(16,-55,1,16)
[A016165(n) for n in range(31)] # G. C. Greubel, Nov 10 2024

a(n) = (11^(n+1) - 5^(n+1))/6. - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
a(n) = 11*a(n-1) + 5^n, a(0)=1. - Vincenzo Librandi, Feb 09 2011
E.g.f.: (1/6)*(11*exp(11*x) - 5*exp(5*x)). - G. C. Greubel, Nov 10 2024

A016166 Expansion of 1/((1-5x)(1-12*x)).

Original entry on oeis.org

1, 17, 229, 2873, 35101, 424337, 5107669, 61370153, 736832461, 8843942657, 106137077509, 1273693758233, 15284569239421, 183416051576177, 2200998722429749, 26412015186735113, 316944334828711981, 3803332780883996897, 45639997185305228389, 547679985297149068793

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..920
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Index entries for linear recurrences with constant coefficients, signature (17,-60).

Cf. A016161.

[(12^(n+1)-5^(n+1))/7 : n in [0..30]]; // Wesley Ivan Hurt, Aug 28 2015

A016166:=n->(12^(n+1)-5^(n+1))/7: seq(A016166(n), n=0..30); # Wesley Ivan Hurt, Aug 28 2015

LinearRecurrence[{17,-60}, {1,17}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
Table[(12^(n+1)-5^(n+1))/7, {n,0,40}] (* Wesley Ivan Hurt, Aug 28 2015 *)

A016166=BinaryRecurrenceSequence(17,-60,1,17)
[A016166(n) for n in range(41)] # G. C. Greubel, Nov 10 2024

G.f.: 1/((1-5*x)*(1-12*x)).
a(n) = (12^(n+1) - 5^(n+1))/7. - Bruno Berselli, Feb 09 2011
a(n) = 12*a(n-1) + 5^n, a(0)=1. - Vincenzo Librandi, Feb 09 2011
a(n) = 17*a(n-1) - 60*a(n-2), n > 1. - Wesley Ivan Hurt, Aug 28 2015
E.g.f.: (1/7)*(12*exp(12*x) - 5*exp(5*x)). - G. C. Greubel, Nov 10 2024

A102728 Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 0, 1, 4, 4, 0, 0, 1, 6, 13, 8, 1, 0, 1, 8, 28, 40, 16, 0, 0, 1, 10, 49, 120, 121, 32, 1, 0, 1, 12, 76, 272, 496, 364, 64, 0, 0, 1, 14, 109, 520, 1441, 2016, 1093, 128, 1, 0, 1, 16, 148, 888, 3376, 7448, 8128, 3280, 256, 0, 0, 1, 18, 193, 1400, 6841

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 07 2005

Consider a 2 X 2 matrix M = [N, 1] / [1, N]. The n-th row of the array contains the values of the non-diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non-diagonal entry + (N-1)^k.) Table:
N: row sequence g.f. cross references.
(1^n-(-1)^n)/2 x/((1+1x)(1-1x)) A000035
(2^n-0^n)/2 x/(1-2x) A000079
(3^n-1^n)/2 x/((1-1x)(1-3x)) A003462
(4^n-2^n)/2 x/((1-2x)(1-4x)) A006516
(7^n-3^n)/2 x/((1-3x)(1-5x)) A005059
(6^n-4^n)/2 x/((1-4x)(1-6x)) A016149
(7^n-5^n)/2 x/((1-5x)(1-7x)) A016161 A081200
(8^n-6^n)/2 x/((1-6x)(1-8x)) A016170 A081201
(9^n-7^n)/2 x/((1-7x)(1-9x)) A016178 A081202
(10^n-8^n)/2 x/((1-8x)(1-10x)) A016186 A081203
(11^n-9^n)/2 x/((1-9x)(1-11x)) A016190
(12^n-10^n)/2 x/((1-10x)(1-12x)) A016196
...
Characteristic polynomial x^2-2nx+(n^2-1) has roots n+-1, so if r(n) denotes a row sequence, r(n+1)/r(n) converges to n+1.
Columns follow polynomials with certain binomial coefficients:
column: polynomial
0
1
2n
3n^2+ 1 (see A056107)
4n^3+ 4n (= 8*A006003(n))
5n^4+ 10n^2+ 1
6n^5+ 20n^3+ 6n
7n^6+ 35n^4+ 21n^2+ 1
8; 8n^7+ 56n^5+ 56n^3+ 8n
9n^8+ 84n^6+126n^4+ 36n^2+ 1
10n^9+ 120n^7+252n^5+120n^3+ 10n
11n^10+165n^8+462n^6+330n^4+ 55n^2+ 1

			Array begins:
0,1,0,1,0,1...
0,1,2,4,8,16...
0,1,4,13,40,121...
0,1,6,28,120,496...
0,1,8,49,272,1441...
...

MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M for(k=0,12, for(i=0,k,print1((MM(2,k-i)^i)[1,2],","))) T(n, k) = ((n+1)^k-(n-1)^k)/2 for(k=0,10, for(i=0,10,print1(T(k,i),","));print()) for(k=0,10, for(i=0,10,print1(((k+1)^i-(k-1)^i)/2,","));print()) for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+1)*x)),i),","));print())

A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8*x)).

Original entry on oeis.org

1, 22, 313, 3666, 38493, 377286, 3529681, 31947322, 282198565, 2447183310, 20920905369, 176852694018, 1481626607917, 12322682753494, 101879323774177, 838170485025354, 6867569457133749, 56077266261254238

Offset: 0

N. J. A. Sloane

From Bruno Berselli, May 09 2013: (Start)
a(n) - 2*a(n-1), for n>0, gives A019928 (after 1);
a(n) - 5*a(n-1), for n>0, gives A016311 (after 1);
a(n) - 7*a(n-1), for n>0, gives A016297 (after 1);
a(n) - 8*a(n-1), for n>0, gives A016296 (after 1);
a(n) - 7*a(n-1) + 10*a(n-2), for n>1, gives A016177 (after 15);
a(n) - 9*a(n-1) + 14*a(n-2), for n>1, gives A016162 (after 13);
a(n) - 10*a(n-1) + 16*a(n-2), for n>1, gives A016161 (after 12);
a(n) - 12*a(n-1) + 35*a(n-2), for n>1, gives A016131 (after 10);
a(n) - 13*a(n-1) + 40*a(n-2), for n>1, gives A016130 (after 9);
a(n) - 15*a(n-1) + 56*a(n-2), for n>1, gives A016127 (after 7);
a(n) - 20*a(n-1) +131*a(n-2) -280*a(n-3), for n>2, gives A000079 (after 4);
a(n) - 17*a(n-1) +86*a(n-2) -112*a(n-3), for n>2, gives A000351 (after 25);
a(n) - 15*a(n-1) +66*a(n-2) -80*a(n-3), for n>2, gives A000420 (after 49);
a(n) - 14*a(n-1) +59*a(n-2) -70*a(n-3), for n>2, gives A001018 (after 64),
and naturally: a(n) - 22*a(n-1) + 171*a(n-2) - 542*a(n-3) + 560*a(n-4), for n>3, gives 0 (see Harvey P. Dale in Formula lines). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (22,-171,542,-560).

Cf. A000079, A000351, A000420, A001018, A016127, A016130, A016131, A016161, A016162, A016177, A016296, A016297, A016311, A019928.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)))); // Bruno Berselli, May 09 2013

CoefficientList[Series[1/((1-2x)(1-5x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {22,-171,542,-560},{1,22,313,3666},30] (* Harvey P. Dale, Jan 29 2013 *)

a(n) = n+=3; (5*8^n-9*7^n+5*5^n-2^n)/90 \\ Charles R Greathouse IV, Oct 03 2016

def A025992(n): return (5*pow(8,n+3)-9*pow(7,n+3)+pow(5,n+4)-pow(2,n+3))//90
print([A025992(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

a(0)=1, a(1)=22, a(2)=313, a(3)=3666, a(n) = 22*a(n-1) - 171*a(n-2) + 542*a(n-3) - 560*a(n-4). - Harvey P. Dale, Jan 29 2013
a(n) = (5*8^(n+3) - 9*7^(n+3) + 5^(n+4) - 2^(n+3))/90. - Yahia Kahloune, May 07 2013
E.g.f.: (1/90)*(-8*exp(2*x) + 625*exp(5*x) - 3087*exp(7*x) + 2560*exp(8*x)). - G. C. Greubel, Dec 27 2024

A105373 Square array by antidiagonals of number of straight lines with n points in a k-dimensional hypercube with n points on each edge.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 28, 8, 1, 1, 120, 49, 10, 1, 1, 496, 272, 76, 12, 1, 1, 2016, 1441, 520, 109, 14, 1, 1, 8128, 7448, 3376, 888, 148, 16, 1, 1, 32640, 37969, 21280, 6841, 1400, 193, 18, 1, 1, 130816, 192032, 131776, 51012, 12496, 2080, 244, 20, 1, 1, 523776

Offset: 1

Henry Bottomley, Apr 02 2005

			Rows start:
  1,  1,   1,   1,    1,     1, ...;
  1,  6,  28, 120,  496,  2016, ...;
  1,  8,  49, 272, 1441,  7448, ...;
  1, 10,  76, 520, 3376, 21280, ...;
  1, 12, 109, 888, 6841, 51012, ...;
  etc.
T(5,3)=109 because in a 5 X 5 X 5 cube there are 25 columns, 25 linear rows in one direction, 25 linear rows in another direction, 5 short diagonals in each of 6 directions and 4 long diagonals; and 3*25 + 6*5 + 4 = 109.

See A102728. Rows essentially include A000012, A006516, A005059, A016149 or A081199, A016161 or A081200, A016170 or A081201, A016178 or A081202 etc. Columns essentially include A000012, A005843, A056107, A105373.

T(1, k)=1. For n>1: T(n, k) = ((n+2)^k-n^k)/2 = (n+2)*T(n, k-1)+n^(k-1) = A102728(k, n+1).

cp's OEIS Frontend

A081200 6th binomial transform of (0,1,0,1,0,1,...), A000035.

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A016166 Expansion of 1/((1-5*x)*(1-12*x)).

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Magma

A016164 Expansion of 1/((1-5x)(1-10*x)).

A016162 Expansion of 1/((1-5x)(1-8*x)).

A016163 Expansion of 1/((1-5x)(1-9*x)).

A016165 Expansion of 1/((1-5x)(1-11*x)).

A016166 Expansion of 1/((1-5x)(1-12*x)).

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