A259239 -id:A259239 - cp's OEIS frontend

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

Karol A. Penson and Katarzyna Gorska, Jun 23 2015

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

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

Cf. A259239.

# 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

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 *)

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

A259367 E.g.f.: exp(x-(1-x^3)^(1/3)+1).

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 161, 911, 3473, 48329, 406241, 2150171, 44216921, 491897693, 3327845249, 90934644711, 1257256962081, 10352273016081, 353351881109313, 5836715156967219, 56621346170765481, 2319460179075419941, 44545835926727113441, 497433851743810193983, 23782590451590763744689

Offset: 0

Karol A. Penson and Katarzyna Gorska, Jun 25 2015

nonn

Cf. A259239,A067622.

nmax = 30; CoefficientList[Series[E^(x-(1-x^3)^(1/3)+1), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) ~ n! * (exp(2) + 2*exp(1/2) * cos((4*Pi*n - 3*sqrt(3))/6)) / (3^(2/3)*Gamma(2/3)*n^(4/3)) * (1 - 3^(5/6)*Gamma(2/3)^2 / (2*Pi*n^(1/3))). - Vaclav Kotesovec, Jun 08 2021

cp's OEIS Frontend

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)).

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