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

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).
Leftmost column is A020557(n) = A000110(2*n), for n >= 0.
Row sums form A326433.

Crossrefs

Cf. A000110, A020557, A108459, A326433, A326434, A326435, A326436, A326437.

Cf. A326601 (central terms).

Sequence in context: A181869 A141510 A219899 * A272304 A266521 A039652

Adjacent sequences: A326597 A326598 A326599 * A326601 A326602 A326603

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 20 2019

Status

approved