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A259286 Triangle of polynomials P(n,y) of order n in y, generated by the extension to the variable y of the e.g.f. of A259239(n), i.e., exp(y*(x-sqrt(1-x^2)+1)).

%I A259286 #37 Jun 28 2018 08:55:00
%S A259286 1,1,1,0,3,1,3,3,6,1,0,15,15,10,1,45,45,60,45,15,1,0,315,315,210,105,
%T A259286 21,1,1575,1575,1890,1365,630,210,28,1,0,14175,14175,9450,4725,1638,
%U A259286 378,36,1,99225,99225,113400,80325,38745,14175,3780,630,45,1
%N A259286 Triangle of polynomials P(n,y) of order n in y, generated by the extension to the variable y of the e.g.f. of A259239(n), i.e., exp(y*(x-sqrt(1-x^2)+1)).
%C A259286 Explicit forms of the polynomials P(n,y) for n=1..6:
%C A259286 P(1,y) = y
%C A259286 P(2,y) = y + y^2
%C A259286 P(3,y) = 3*y^2 + y^3
%C A259286 P(4,y) = 3*y + 3*y^2 + 6*y^3 + 1*y^4
%C A259286 P(5,y) = 15*y^2 + 15*y^3 + 10*y^4 + 1*y^5
%C A259286 P(6,y) = 45*y + 45*y^2 + 60*y^3 + 45*y^4 + 15*y^5 + 1*y^6;
%C A259286 Sum(k=1..n, P(k,1) ) = A259239(n).
%C A259286 Also the Bell transform of the sequence "a(n)=n*doublefactorial(n-2)^2 if n is odd else 0^n". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%e A259286 Triangle begins:
%e A259286   1;
%e A259286   1,  1;
%e A259286   0,  3,  1;
%e A259286   3,  3,  6,  1;
%e A259286   0, 15, 15, 10,  1;
%p A259286 # The function BellMatrix is defined in A264428.
%p A259286 # Adds (1,0,0,0, ..) as column 0.
%p A259286 BellMatrix(n -> `if`(n::even,0^n,n*doublefactorial(n-2)^2), 9); # _Peter Luschny_, Jan 29 2016
%t A259286 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A259286 B = BellMatrix[Function[n, Which[n==0, 1, EvenQ[n], 0, True, n*(n-2)!!^2]], rows = 12];
%t A259286 Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A259286 (PARI) row(n) = x='x+O('x^(n+1));polcoeff(serlaplace(exp(y*(x-sqrt(1-x^2)+1))), n, 'x);
%o A259286 tabl(nn) = for (n=1, nn, print(Vecrev(row(n)/y))) \\ _Michel Marcus_, Jun 23 2015
%Y A259286 Cf. A259239.
%K A259286 nonn,tabl
%O A259286 1,5
%A A259286 _Karol A. Penson_ and Katarzyna Gorska, Jun 23 2015