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A137378
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.
1
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, 1, 0, 25200, 0, -7560, 0, 210, 0, -1, 0, 0, -282240, 0, 23520, 0, -336, 0, 1, 0, -5080320, 0, 1693440, 0, -60480, 0, 504, 0, -1, 0, 0, 76204800, 0, -7257600, 0, 136080, 0, -720, 0, 1
OFFSET
0,8
COMMENTS
Row sums are: 1, -1, 1, 5, -23, -181, 1681, 17849, -259055, -3446857, 69082561,..
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
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
LINKS
Ömür Kıvanç Kürkçü, A new numerical method for solving delay integral equations with variable bounds by using generalized Mott polynomials, Anadolu University Journal of Science and Technology A, Applied Sciences and Engineering (2018) Vol. 19, No. 4, 264-277.
N. F. Mott, The polarisation of electrons by double scattering, Proc. R. Soc. Lond. A 135 (827) (1932), p 442.
Eric Weisstein's MathWorld, Mott Polynomial
Wikipedia, Mott Polynomials
FORMULA
p(x) = Exp[x*(1 - Sqrt[1 + t^2])/t]; weights 2^n*n!;
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]
EXAMPLE
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, 1;
0, 25200, 0, -7560, 0, 210, 0, -1;
0, 0, -282240, 0, 23520, 0, -336, 0, 1;
0, -5080320, 0, 1693440, 0, -60480, 0, 504, 0, -1;
0, 0, 76204800, 0, -7257600, 0, 136080, 0, -720, 0, 1;
MAPLE
# The function BellMatrix is defined in A264428.
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
A137378 := proc(n, k)
local m ;
if n =0 and k =0 then
1;
elif type(n+k, 'odd') then
0;
else
m := (n+k)/2 ;
(-1)^m*k*binomial(n-1, m-1)/m/k! ;
%*n! ;
end if;
end proc: # R. J. Mathar, Nov 17 2018
MATHEMATICA
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]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[OddQ[n], 0, (-1)^(1 + n/2)*(n + 1)/(n/2 + 1)*(n!/(n/2)!)^2]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
PROG
(Maxima)
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);
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
CROSSREFS
Sequence in context: A240315 A339431 A256041 * A333275 A293071 A084680
KEYWORD
tabl,sign
AUTHOR
Roger L. Bagula, Apr 09 2008
STATUS
approved