A081199 -id:A081199 - cp's OEIS frontend

A005059 a(n) = (5^n - 3^n)/2.

0, 1, 8, 49, 272, 1441, 7448, 37969, 192032, 966721, 4853288, 24325489, 121804592, 609554401, 3049366328, 15251614609, 76272421952, 381405156481, 1907154922568, 9536162033329, 47681972428112, 238413348924961, 1192077204978008, 5960417405949649, 29802181172927072

Offset: 0

N. J. A. Sloane

Number of lines passing through 3 points of an n-dimensional grid of points of side 3. - David W. Wilson, c. 1999

a(n) is also the total number of words of length n, over an alphabet of five letters, one of them appearing an odd number of times. See the Lekraj Beedassy, Jul 22 2003, comment under A006516 (4-letter words), and the Balakrishnan reference there. See A003462 for the analogous 3-letter words problem. - Wolfdieter Lang, Jul 16 2017

			For the fifth formula: a(4) = 1*125 + 3*25 + 9*5 + 27*1 = 272. [_Bruno Berselli_, Aug 07 2013]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. A. Alekseyev and T. Berger, Solving the Tower of Hanoi with Random Moves. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
Index entries for linear recurrences with constant coefficients, signature (8,-15).

Cf. A081199 (binomial transform), A006516 (inverse binomial transform, and special 4-letter words), A003462 (special 3-letter words).

[(5^n - 3^n)/2: n in [0..30] ]; // Vincenzo Librandi, Aug 19 2011

A005059:=n->(5^n-3^n)/2: seq(A005059(n), n=0..30); # Wesley Ivan Hurt, Nov 18 2014

Join[{a=0,b=1},Table[c=8*b-15*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8, -15}, {0, 1}, 50] (* Sture Sjöstedt, Oct 17 2012 *)
Table[(5^n - 3^n)/2, {n, 0, 23}] (* Michael De Vlieger, Jul 16 2017 *)

a(n)=(5^n-3^n)/2 \\ Charles R Greathouse IV, Jun 11 2013

[lucas_number1(n,8,15) for n in range(0, 21)] # Zerinvary Lajos, Apr 23 2009

From Paul Barry, Mar 03 2003: (Start)
a(n) = 8*a(n-1) - 15*a(n-2).
G.f.: x/((1-3*x)*(1-5*x)). (End)
a(n) = Sum_{k=1..n} 2^(k-1)*3^(n-k)*binomial(n,k). - Zerinvary Lajos, Sep 24 2006
a(n) = (r^n-s^n)/(r-s) with r=5 and s=3. - Sture Sjöstedt, Oct 17 2012
a(n) = Sum_{k=0..n-1} 3^k*5^(n-k-1) for n>0, a(0)=0. - Bruno Berselli, Aug 07 2013
E.g.f.: exp(3*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jul 23 2024

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

Original entry on oeis.org

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

A016149 Expansion of g.f. 1/((1 - 4x)(1 - 6*x)).

Original entry on oeis.org

1, 10, 76, 520, 3376, 21280, 131776, 807040, 4907776, 29708800, 179301376, 1080002560, 6496792576, 39047864320, 234555621376, 1408407470080, 8454739787776, 50745618595840, 304542431051776, 1827529464217600, 10966276296933376, 65802055828111360, 394829927154712576

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-24).

Cf. A081199, A080961.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-6*x)))); // Vincenzo Librandi, Jun 24 2013

seq(add(2^(2*n-k)*binomial(n,k)/2,k=1..n),n=1..19); # Zerinvary Lajos, Apr 18 2009

Join[{a=1,b=10},Table[c=10*b-24*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
LinearRecurrence[{10,-24},{1,10},30]  (* or *) CoefficientList[ Series[ 1/(1-10 x+24 x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2011 *)

Vec(1/((1-4*x)*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

[lucas_number1(n,10,24) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009

From Barry E. Williams, Jan 13 2000: (Start)
a(n) = 10*a(n-1) - 24*a(n-2) for n>1, a(0)=1.
a(n) = ((6^(n+1)) - 4^(n+1))/2. (End)
a(n) = A081199(n+1). Binomial transform of A080961. - R. J. Mathar, Sep 18 2008
a(n) = Sum_{k=0..n} 6^k*4^(n-k). - Bruno Berselli, Aug 07 2013
E.g.f.: exp(4*x)*(3*exp(2*x) - 2). - Stefano Spezia, Jul 23 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).

A248337 a(n) = 6^n - 4^n.

Original entry on oeis.org

0, 2, 20, 152, 1040, 6752, 42560, 263552, 1614080, 9815552, 59417600, 358602752, 2160005120, 12993585152, 78095728640, 469111242752, 2816814940160, 16909479575552, 101491237191680, 609084862103552, 3655058928435200, 21932552593866752, 131604111656222720, 789659854309425152, 4738099863344906240, 28429162130022858752

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-24).

Cf. sequences of the form k^n - 4^n: -A000302 (k=0), -A024036 (k=1), -A020522 (k=2), -A005061 (k=3), A005060 (k=5), this sequence (k=6), A190542 (k=7), A059409 (k=8), A118004 (k=9), A248338 (k=10), A139742 (k=11), 8*A016159 (k=12).

Cf. A000079, A000302, A000400, A001047, A081199.

Magma
```
[6^n-4^n: n in [0..30]];
```

Table[6^n - 4^n, {n,0,30}]
CoefficientList[Series[(2 x)/((1-4 x)(1-6 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-24},{0,2},30] (* Harvey P. Dale, Aug 18 2024 *)

vector(20,n,6^(n-1)-4^(n-1)) \\ Derek Orr, Oct 05 2014

A248337=BinaryRecurrenceSequence(10,-24,0,2)
[A248337(n) for n in range(31)] # G. C. Greubel, Nov 11 2024

G.f.: 2*x/((1-4*x)*(1-6*x)).
a(n) = 10*a(n-1) - 24*a(n-2).
a(n) = 2^n*(3^n-2^n) =  A000079(n) * A001047(n) = A000400(n) - A000302(n).
a(n) = 2*A081199(n). - Bruno Berselli, Oct 05 2014
E.g.f.: 2*exp(5*x)*sinh(x). - G. C. Greubel, Nov 11 2024

More terms added by G. C. Greubel, Nov 11 2024

A327318 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 1 and s = 1/2.

Original entry on oeis.org

1, 3, 4, 7, 18, 12, 15, 56, 72, 32, 31, 150, 280, 240, 80, 63, 372, 900, 1120, 720, 192, 127, 882, 2604, 4200, 3920, 2016, 448, 255, 2032, 7056, 13888, 16800, 12544, 5376, 1024, 511, 4590, 18288, 42336, 62496, 60480, 37632, 13824, 2304, 1023, 10220, 45900

Offset: 1

Clark Kimberling, Nov 08 2019

p(x,n) is a strong divisibility sequence of polynomials. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			First six rows:
   1;
   3,   4;
   7,  18,  12;
  15,  56,  72,   32;
  31, 150, 280,  240,  80;
  63, 372, 900, 1120, 720, 192;
The first six polynomials, not factored:
1, 3 + 4 x, 7 + 18 x + 12 x^2, 15 + 56 x + 72 x^2 + 32 x^3, 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, 63 + 372 x + 900 x^2 + 1120 x^3 + 720 x^4 + 192 x^5.
The first six polynomials, factored:
1, 3 + 4 x, 7 + 18 x + 12 x^2, (3 + 4 x) (5 + 12 x + 8 x^2), 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, (3 + 4 x) (3 + 6 x + 4 x^2) (7 + 18 x + 12 x^2).

Cf. A327316, A327317, A000225 (x = 0), A005061 (x = 1), A081199 (x = 1/2).

r = 1; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327318 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327318 sequence *)

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

A005059 a(n) = (5^n - 3^n)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A016149 Expansion of g.f. 1/((1 - 4*x)*(1 - 6*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A248337 a(n) = 6^n - 4^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A327318 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 1 and s = 1/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

A016149 Expansion of g.f. 1/((1 - 4x)(1 - 6*x)).