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)).

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, 21, 1, 1575, 1575, 1890, 1365, 630, 210, 28, 1, 0, 14175, 14175, 9450, 4725, 1638, 378, 36, 1, 99225, 99225, 113400, 80325, 38745, 14175, 3780, 630, 45, 1

Offset

1,5

Comments

Explicit forms of the polynomials P(n,y) for n=1..6:

P(1,y) = y

P(2,y) = y + y^2

P(3,y) = 3*y^2 + y^3

P(4,y) = 3*y + 3*y^2 + 6*y^3 + 1*y^4

P(5,y) = 15*y^2 + 15*y^3 + 10*y^4 + 1*y^5

P(6,y) = 45*y + 45*y^2 + 60*y^3 + 45*y^4 + 15*y^5 + 1*y^6;

Sum(k=1..n, P(k,1) ) = A259239(n).

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

Links

Table of n, a(n) for n=1..55.

Example

Triangle begins:
  1;
  1,  1;
  0,  3,  1;
  3,  3,  6,  1;
  0, 15, 15, 10,  1;

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n::even, 0^n, n*doublefactorial(n-2)^2), 9); # Peter Luschny, Jan 29 2016

Mathematica

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, Which[n==0, 1, EvenQ[n], 0, True, n*(n-2)!!^2]], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Prog

(PARI) row(n) = x='x+O('x^(n+1)); polcoeff(serlaplace(exp(y*(x-sqrt(1-x^2)+1))), n, 'x);
tabl(nn) = for (n=1, nn, print(Vecrev(row(n)/y))) \\ Michel Marcus, Jun 23 2015

Crossrefs

Cf. A259239.

Sequence in context: A165027 A342341 A078555 * A087891 A005885 A333334

Adjacent sequences: A259283 A259284 A259285 * A259287 A259288 A259289

Keyword

nonn,tabl

Author

Karol A. Penson and Katarzyna Gorska, Jun 23 2015

Status

approved