OFFSET
0,2
FORMULA
a(n) = [x^(2*n)*y^n/n!] exp(-1-y) * Sum_{m>=0} (exp(m*x) + y)^m / m!.
a(n) = [x^(2*n)*y^n/n!] exp(-1-y) * Sum_{m>=0} exp(m^2*x) * exp( y*exp(m*x) ) / m!.
EXAMPLE
E.g.f. of A326600 begins
F(x,y) = 1 + (2 + y)*x + (15 + 12*y + 2*y^2)*x^2/2! + (203 + 206*y + 60*y^2 + 5*y^3)*x^3/3! + (4140 + 4949*y + 1947*y^2 + 298*y^3 + 15*y^4)*x^4/4! + (115975 + 156972*y + 75595*y^2 + 16160*y^3 + 1535*y^4 + 52*y^5)*x^5/5! + (4213597 + 6301550*y + 3528368*y^2 + 945360*y^3 + 127915*y^4 + 8307*y^5 + 203*y^6)*x^6/6! + (190899322 + 310279615*y + 195764198*y^2 + 62079052*y^3 + 10690645*y^4 + 1001567*y^5 + 47397*y^6 + 877*y^7)*x^7/7! + (10480142147 + 18293310174*y + 12735957930*y^2 + 4614975428*y^3 + 952279230*y^4 + 114741060*y^5 + 7901236*y^6 + 285096*y^7 + 4140*y^8)*x^8/8! + ...
such that
F(x,y) = exp(-1-y) * (1 + (exp(x) + y) + (exp(2*x) + y)^2/2! + (exp(3*x) + y)^3/3! + (exp(4*x) + y)^4/4! + (exp(5*x) + y)^5/5! + (exp(6*x) + y)^6/6! + ...)
also
F(x,y) = exp(-1-y) * (exp(y) + exp(x)*exp(y*exp(x)) + exp(4*x)*exp(y*exp(2*x))/2! + exp(9*x)*exp(y*exp(3*x))/3! + exp(16*x)*exp(y*exp(4*x))/4! + exp(25*x)*exp(y*exp(5*x))/5! + exp(36*x)*exp(y*exp(6*x))/6! + ...).
Triangle A326600 of coefficients of x^n*y^k/n! in F(x,y) begins:
[1],
[2, 1],
[15, 12, 2],
[203, 206, 60, 5],
[4140, 4949, 1947, 298, 15],
[115975, 156972, 75595, 16160, 1535, 52],
[4213597, 6301550, 3528368, 945360, 127915, 8307, 203],
[190899322, 310279615, 195764198, 62079052, 10690645, 1001567, 47397, 877],
[10480142147, 18293310174, 12735957930, 4614975428, 952279230, 114741060, 7901236, 285096, 4140], ...
in which the central terms form this sequence.
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 22 2019
STATUS
approved