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%I #27 Dec 06 2023 12:26:20
%S 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,
%T 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,
%U 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
%N Polynomials with e.g.f. exp(x*t)/csch(t), triangle of coefficients read by rows.
%C 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
%C 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
%D Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.
%H Luca Onnis, <a href="/A162590/a162590.gif">Animation of the Ehrenfest model</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrenfest_model">Ehrenfest model</a>.
%F p_n(x) = Sum_{k=0..n} (k mod 2)*binomial(n,k)*x^(n-k).
%F 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!) + ...
%F The 'co'-polynomials with generating function exp(x*t)*sech(t) are the Swiss-Knife polynomials (A153641).
%e Triangle begins:
%e 0
%e 1, 0
%e 0, 2, 0
%e 1, 0, 3, 0
%e 0, 4, 0, 4, 0
%e 1, 0, 10, 0, 5, 0
%e 0, 6, 0, 20, 0, 6, 0
%e 1, 0, 21, 0, 35, 0, 7, 0
%e ...
%e p[0](x) = 0;
%e p[1](x) = 1
%e p[2](x) = 2*x
%e p[3](x) = 3*x^2 + 1
%e p[4](x) = 4*x^3 + 4*x
%e p[5](x) = 5*x^4 + 10*x^2 + 1
%e p[6](x) = 6*x^5 + 20*x^3 + 6*x
%e p[7](x) = 7*x^6 + 35*x^4 + 21*x^2 + 1
%e p[8](x) = 8*x^7 + 56*x^5 + 56*x^3 + 8*x
%e .
%e Cf. the triangle of odd-numbered terms in rows of Pascal's triangle (A034867).
%e p[n] (k), n=0,1,...
%e k=0: 0, 1, 0, 1, 0, 1, ... A000035, (A059841)
%e k=1: 0, 1, 2, 4, 8, 16, ... A131577, (A000079)
%e k=2: 0, 1, 4, 13, 40, 121, ... A003462
%e k=3: 0, 1, 6, 28, 120, 496, ... A006516
%e k=4: 0, 1, 8, 49, 272, 1441, ... A005059
%e k=5: 0, 1, 10, 76, 520, 3376, ... A081199, (A016149)
%e k=6: 0, 1, 12, 109, 888, 6841, ... A081200, (A016161)
%e k=7: 0, 1, 14, 148, 1400, 12496, ... A081201, (A016170)
%e k=8: 0, 1, 16, 193, 2080, 21121, ... A081202, (A016178)
%e k=9: 0, 1, 18, 244, 2952, 33616, ... A081203, (A016186)
%e k=10: 0, 1, 20, 301, 4040, 51001, ... ......., (A016190)
%e .
%e p[n] (k), k=0,1,...
%e p[0]: 0, 0, 0, 0, 0, 0, ... A000004
%e p[1]: 1, 1, 1, 1, 1, 1, ... A000012
%e p[2]: 0, 2, 4, 6, 8, 10, ... A005843
%e p[3]: 1, 4, 13, 28, 49, 76, ... A056107
%e p[4]: 0, 8, 40, 120, 272, 520, ... A105374
%e p[5]: 1, 16, 121, 496, 1441, 3376, ...
%e p[6]: 0, 32, 364, 2016, 7448, 21280, ...
%p # Polynomials: p_n(x)
%p p := proc(n,x) local k;
%p pow := (n,k) -> `if`(n=0 and k=0,1,n^k);
%p add((k mod 2)*binomial(n,k)*pow(x,n-k),k=0..n) end;
%p # Coefficients: a(n)
%p seq(print(seq(coeff(i!*coeff(series(exp(x*t)/csch(t), t,16),t,i),x,n), n=0..i)), i=0..8);
%t 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 *)
%t n = 15; "n-th row"
%t mat = Table[Table[0, {j, 1, n + 1}], {i, 1, n + 1}];
%t mat[[1, 2]] = 1;
%t mat[[n + 1, n]] = 1;
%t For[i = 2, i <= n, i++, mat[[i, i - 1]] = (i - 1)/n ];
%t For[i = 2, i <= n, i++, mat[[i, i + 1]] = (n - i + 1)/n];
%t mat // MatrixForm;
%t P2 = Dot[mat, mat];
%t R1 = Simplify[
%t Eigenvectors[Transpose[P2]][[1]]/
%t Total[Eigenvectors[Transpose[P2]][[1]]]]
%t R2 = Table[Dot[R1, Transpose[mat][[k]]], {k, 1, n + 1}]
%t even = R1*2^(n - 1) (* _Luca Onnis_, Oct 29 2023_ *)
%Y Cf. A119467.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jul 07 2009
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