Limit in Triangles for Generalized Bell Numbers, Factorials

[Paul D. Hanna]

Seqfans,
     Inspired by one of Vladeta Jovovic's formulas involving a 
generalization of W(x) = LambertW(-x)/(-x):
"W(x,q) = Sum_{n>=0} w(n,q)*x^n, where 
w(n+1,q) = 1/(n+1)*Sum_{i=0..n} (i+1)*w(i,q)*w(n-i,q)*q^i."
I explored some variants and found some interesting sequence limits. 
  
The variants are: 
(1) Generalized Bell numbers (new triangle A126347):
B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1.
and 
(2) Generalized Factorials (new triangle A126470): 
F(n,q) = Sum_{k=0..n-1} C(n-1,k)*F(k,q)*F(n-k-1,q)*q^k for n>0, with
F(0,q) = 1.
 
What is interesting is that the reversed rows of the triangles tend to a
limit.
Below I copy the sequences that are the limit of the row reversals. 
 
Perhaps someone can find a recurrence or e.g.f. for these limits? 
 
Happy New Year to all, 
       Paul 
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(1) Generalized Bell numbers - limit of row reversal. 
 
New sequence A126348: 
Limit of reversed rows of triangle A126347, in which row sums equal Bell
numbers (A000110).
 
1,1,2,4,7,12,20,33,53,84,131,202,308,465,695,1030,1514,2209,3201,4609,
6596,9386,13284,18705,26211,36561,50776,70226,96742,132765,181540,247369,
335940,454756,613689,825698,1107755,1482038,1977465,2631664,3493496,
 
In triangle A126347, row n lists coefficients of q in B(n,q) that
satisfies: 
B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1.
 
Row functions B(n,q) in triangle A126347 begin: 
B(0,q) = B(1,q) = 1 ; 
B(1,q) = 1 + q ; 
B(2,q) = 1 + 2*q + q^2 + q^3 ;
B(3,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.
Triangle A126347 begins:
1;
1;
1, 1;
1, 2, 1, 1;
1, 3, 3, 4, 2, 1, 1;
1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;
1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;
1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4,
2, 1, 1; ...
 
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(2) Generalized Factorials - limit of row reversal.
 
New sequence A126471: 
Limit of reversed rows of triangle A126470, in which row sums equal the
factorials.
 
1,1,3,5,12,17,39,58,108,170,310,449,791,1181,1960,2915,4668,6822,10842,
15818,24254,35061,53213,76061,113822,162631,238660,337764,491319,690530,
994390,1391968,1982724,2757196,3896450,5382342,7546547,10384787,14450140,
 
In triangle A126470, row n lists coefficients of q in F(n,q) that
satisfies: 
F(n,q) = Sum_{k=0..n-1} C(n-1,k)*F(k,q)*F(n-k-1,q)*q^k for n>0, with
F(0,q) = 1.
 
Row functions F(n,q) of triangle A126470 begin: 
F(0,q) = F(1,q) = 1 ; 
F(1,q) = 1 + q ; 
F(2,q) = 1 + 3*q + q^2 + q^3 ;
F(3,q) = 1 + 6*q + 7*q^2 + 5*q^3 + 3*q^4 + q^5 + q^6.
Triangle A126470 begins:
1;
1;
1, 1;
1, 3, 1, 1;
1, 6, 7, 5, 3, 1, 1;
1, 10, 25, 25, 26, 11, 12, 5, 3, 1, 1;
1, 15, 65, 110, 136, 117, 92, 70, 43, 32, 17, 12, 5, 3, 1, 1;
1, 21, 140, 385, 616, 784, 694, 687, 478, 411, 255, 222, 127, 91, 50, 39,
17, 12, 5, 3, 1, 1; ...
 
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END.