A081202 -id:A081202 - cp's OEIS frontend

A081201 7th binomial transform of (0,1,0,1,0,1,....), A000035.

0, 1, 14, 148, 1400, 12496, 107744, 908608, 7548800, 62070016, 506637824, 4113568768, 33271347200, 268347559936, 2159841173504, 17357093552128, 139326933401600, 1117436577120256, 8956419276406784, 71752914167922688, 574632673083392000, 4600717543107198976

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A081200.
Conjecture (verified up to a(8)): Number of collinear 6-tuples of points in a 6 X 6 X 6 X... n-dimensional cubic grid. [R. H. Hardin, May 23 2010]
From Wolfdieter Lang, Jul 17 2017: (Start)
For a combinatorial interpretation of a(n) with special 8-letter words of length n see the comment in A081200 on the 7-letter analog.
The binomial transform of {a(n)}_{n >= 0} is A081202, the 9-letter analog.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (14,-48).

Cf. A000035, A016169, A081200, A081202.

Apart from offset same as A016170.

[(8^n-6^n)/2: n in [0..30]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x/((1 - 6 x) (1 - 8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{14,-48},{0,1},30] (* Harvey P. Dale, Oct 24 2022 *)

A081201=BinaryRecurrenceSequence(14,-48,0,1)
[A081201(n) for n in range(41)] # G. C. Greubel, Nov 10 2024

a(n) = 14*a(n-1) - 48*a(n-2) with n>1, a(0)=0, a(1)=1.
G.f.: x/((1-6*x)*(1-8*x)).
a(n) = (1/2)*(8^n - 6^n).
E.g.f.: exp(7*x)*sinh(x). - G. C. Greubel, Nov 10 2024

Name clarified by Pontus von Brömssen, Nov 11 2020

A081203 9th binomial transform of (0,1,0,1,0,1,.....), A000035.

Original entry on oeis.org

0, 1, 18, 244, 2952, 33616, 368928, 3951424, 41611392, 432891136, 4463129088, 45705032704, 465640261632, 4725122093056, 47800976744448, 482407813955584, 4859262511644672, 48874100093157376, 490992800745259008

Offset: 0

Paul Barry, Mar 11 2003

Binomial transform of A081202.
From Wolfdieter Lang, Jul 17 2017: (Start)
For a combinatorial interpretation of a(n) with special 10-letter words of length n see the comment in A081200 on the 7-letter analog.
The binomial transform of {a(n)}_{n >= 0} is {0, A016190}, the 11-letter analog.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-80).

Apart from the first term, identical to A016186.

Cf. A000035, A016190, A081202, A081200, A081201, A081202.

[10^n/2 - 8^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x / ((1 - 8 x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{18,-80},{0,1},20] (* Harvey P. Dale, Aug 05 2018 *)

a(n) = 18*a(n-1)-80*a(n-2), a(0)=0, a(1)=1.
G.f.: x/((1-8*x)*(1-10*x)).
a(n) = 10^n/2 - 8^n/2.

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).

A016178 Expansion of g.f. 1/((1 - 7x)*(1 - 9x)).

Original entry on oeis.org

1, 16, 193, 2080, 21121, 206896, 1979713, 18640960, 173533441, 1602154576, 14701866433, 134294124640, 1222488408961, 11099284691056, 100571785292353, 909893629141120, 8222275592839681, 74233110849544336, 669726411243809473, 6038936596379658400, 54430221633714537601

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (16,-63).

Cf. A081034, A081202.

Join[{a=1,b=16},Table[c=16*b-63*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
CoefficientList[Series[1/((1-7x)(1-9x)),{x,0,30}],x] (* or *) LinearRecurrence[ {16,-63},{1,16},30] (* Harvey P. Dale, Mar 11 2013 *)

a(n)=(9^n++-7^n)/2 \\ Charles R Greathouse IV, Sep 24 2012

a(n) = (9^(n+1) - 7^(n+1))/2 = A081202(n+1). Binomial transform of A081034. - R. J. Mathar, Sep 18 2008
From Vincenzo Librandi, Feb 09 2011: (Start)
a(n) = 9*a(n-1) + 7^n, with a(0)=1.
a(n) = 16*a(n-1) - 63*a(n-2), n >= 2. (End)
E.g.f.: exp(7*x)*(9*exp(2*x) - 7)/2. - Stefano Spezia, Jul 23 2024

a(18)-a(20) from Stefano Spezia, Jul 23 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())

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).

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A081201 7th binomial transform of (0,1,0,1,0,1,....), A000035.

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A081203 9th binomial transform of (0,1,0,1,0,1,.....), A000035.

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

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A016178 Expansion of g.f. 1/((1 - 7x)*(1 - 9x)).

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

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

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A273010 Numbers n such that (9^n - 7^n)/2 is prime.

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