A137378 - OEIS

%I #31 Feb 11 2020 22:02:00

%S 1,0,-1,0,0,1,0,6,0,-1,0,0,-24,0,1,0,-240,0,60,0,-1,0,0,1800,0,-120,0,

%T 1,0,25200,0,-7560,0,210,0,-1,0,0,-282240,0,23520,0,-336,0,1,0,

%U -5080320,0,1693440,0,-60480,0,504,0,-1,0,0,76204800,0,-7257600,0,136080,0,-720,0,1

%N Triangle of the coefficient [x^k] of the polynomial 2^n*s_n(x) generated by exp(x*(1 - sqrt(1+t^2))/t) = sum_{n>=0} s_n(x)*t^k/k! in row n, column k.

%C Row sums are: 1, -1, 1, 5, -23, -181, 1681, 17849, -259055, -3446857, 69082561,..

%C Weisstein uses the nomenclature "Mott Polynomial" for s_n(x), although his definition differs from Mott's by signs. - _R. J. Mathar_, Oct 03 2013

%C Also the Bell transform of the sequence defined below in the Maple program. For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016

%H Ömür Kıvanç Kürkçü, <a href="https://doi.org/10.18038/aubtda.409056">A new numerical method for solving delay integral equations with variable bounds by using generalized Mott polynomials</a>, Anadolu University Journal of Science and Technology A, Applied Sciences and Engineering (2018) Vol. 19, No. 4, 264-277.

%H Ömür Kıvanç Kürkçü, <a href="https://doi.org/10.1007/s40096-019-00314-8">A numerical method with a control parameter for integro-differential delay equations with state-dependent bounds via generalized Mott polynomial</a>, Mathematical Sciences (2019).

%H N. F. Mott, <a href="http://www.jstor.org/stable/95868">The polarisation of electrons by double scattering</a>, Proc. R. Soc. Lond. A 135 (827) (1932), p 442.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/MottPolynomial.html">Mott Polynomial</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Mott_polynomial">Mott Polynomials</a>

%F p(x) = Exp[x*(1 - Sqrt[1 + t^2])/t]; weights 2^n*n!;

%F M(n,x) = n!/2^n *sum_{m=floor((n+1)/2)..n} ((-1)^m *(2*m-n) *binomial(n-1,m-1) *x^(2*m-n))/(m*(2*m-n)!). [_Dmitry Kruchinin_ Mar 24 2013]

%e 1;

%e 0, -1;

%e 0, 0, 1;

%e 0, 6, 0, -1;

%e 0, 0, -24, 0, 1;

%e 0, -240,0, 60, 0, -1;

%e 0, 0, 1800, 0, -120, 0, 1;

%e 0, 25200, 0, -7560, 0, 210, 0, -1;

%e 0, 0, -282240, 0, 23520, 0, -336, 0, 1;

%e 0, -5080320, 0, 1693440, 0, -60480, 0, 504, 0, -1;

%e 0, 0, 76204800, 0, -7257600, 0, 136080, 0, -720, 0, 1;

%p # The function BellMatrix is defined in A264428.

%p BellMatrix(n -> `if`(n::odd,0,(-1)^(1+n/2)*(n+1)/(n/2+1)*(n!/(n/2)!)^2), 9); # _Peter Luschny_, Jan 27 2016

%p A137378 := proc(n,k)

%p     local m ;

%p     if n =0 and k =0 then

%p         1;

%p     elif type(n+k,'odd') then

%p         0;

%p     else

%p         m := (n+k)/2 ;

%p         (-1)^m*k*binomial(n-1,m-1)/m/k! ;

%p         %*n! ;

%p     end if;

%p end proc: # _R. J. Mathar_, Nov 17 2018

%t p[t_] = Exp[x*(1 - Sqrt[1 + t^2])/t]; Table[ ExpandAll[2^(n)*n! * SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[2^(n)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

%t (* Second program: *)

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, If[OddQ[n], 0, (-1)^(1 + n/2)*(n + 1)/(n/2 + 1)*(n!/(n/2)!)^2]], rows = 12];

%t Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%o (Maxima)

%o M(n):=n!*sum(((2*m-n)*(-1)^(m)*binomial(n-1,m-1)*x^(2*m-n)/((2*m-n)!*(m))),m,floor((n+1)/2),n);

%o for n:0 thru 7  do if n=0 then print([1]) else (LL:makelist(coeff(ratsimp(M(n)),x,k),k,0,n), print(LL)); // _Dmitry Kruchinin_, Mar 24 2013

%K tabl,sign

%O 0,8

%A _Roger L. Bagula_, Apr 09 2008