A326600 - OEIS

A326600

E.g.f.: A(x,y) = exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!, where A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} T(n,k)*y^k, as a triangle of coefficients T(n,k) read by rows.

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, 682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147, 51724158235372, 101557600812015, 82635818516305, 36672690416280, 9831937482310, 1665456655065, 180791918475, 12443391060, 520878315, 12004575, 115975

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OFFSET

0,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..495 (first 30 rows of this triangle).

FORMULA

E.g.f.: exp(-1-y) * Sum_{n>=0} (exp(n*x) + y)^n / n!.

E.g.f.: exp(-1-y) * Sum_{n>=0} exp(n^2*x) * exp( y*exp(n*x) ) / n!.

FORMULAS FOR TERMS.

T(n,n) = A000110(n) for n >= 0, where A000110 is the Bell numbers.

T(n,0) = A000110(2*n) for n >= 0, where A000110 is the Bell numbers.

Sum_{k=0..n} T(n,k) * (-1)^k = A108459(n) for n >= 0.

Sum_{k=0..n} T(n,k) = A326433(n) for n >= 0.

Sum_{k=0..n} T(n,k) * 2^k = A326434(n) for n >= 0.

Sum_{k=0..n} T(n,k) * 3^k = A326435(n) for n >= 0.

Sum_{k=0..n} T(n,k) * 4^k = A326436(n) for n >= 0.

EXAMPLE

E.g.f.: A(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! + (682076806159 + 1267153412532*y + 959061013824*y^2 + 387848415927*y^3 + 92381300277*y^4 + 13455280629*y^5 + 1200540180*y^6 + 63424134*y^7 + 1805067*y^8 + 21147*y^9)*x^9/9! + (51724158235372 + 101557600812015*y + 82635818516305*y^2 + 36672690416280*y^3 + 9831937482310*y^4 + 1665456655065*y^5 + 180791918475*y^6 + 12443391060*y^7 + 520878315*y^8 + 12004575*y^9 + 115975*y^10)*x^10/10! + ...

such that

A(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

A(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! + ...).

This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(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],

[682076806159, 1267153412532, 959061013824, 387848415927, 92381300277, 13455280629, 1200540180, 63424134, 1805067, 21147], ...

Main diagonal is A000110 (Bell numbers).