OFFSET
1,2
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998
FORMULA
E.g.f. (with constant term 1 included): A(x,t) = [ 1/x*Revert( x*(1 - x)^t ) ]^(1/t) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + n,n)*x^n = 1 + x + (2 + 2*t)*x^2/2! + (2 + 3*t)*(3 + 3*t)*x^3/3! + (2 + 4*t)*(3 + 4*t)*(4 + 4*t)*x^4/4! + ..., where Revert denotes the series reversion operator with respect to x.
In the notation of the Bala link, A(x,t) = I^t(1/(1 - x)) where I^t is a fractional inversion operator.
A(x,t) = B_(1+t)(x), where B_t(x) is the e.g.f. for A251592 and is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.
A(x,t)^m = Sum_{n >= 0} m/(n*t + m)*binomial(n*t + n + m - 1,n)*x^n = 1 + m*x + m*(2*t + m + 1)*x^2/2! + m*(3*t + m + 1)*(3*t + m + 2)*x^3/3! + m*(4*t + m + 1)*(4*t + m + 2)*(4*t + m + 3)*x^4/4! + ....
A(x,t)^t = 1 + t*x + t(1 + 3*t)*x^2/2! + t*(1 + 4*t)*(2 + 4*t)*x^3/3! + t*(1 + 5*t)*(2 + 5*t)*(3 + 5*t)*x^4/4! + ... is the e.g.f for A220883 with an extra constant term 1 and an extra factor of t included.
t*log( A(x,t) ) = t*x + t*(1 + 2*t)*x^2/2! + t*(1 + 3*t)*(2 + 3*t)*x^3/3! + t*(1 + 4*t)*(2 + 4*t)*(3 + 4*t)*x^4/4! + ... is the e.g.f for A056856.
For n = 1,2,3,..., the sequence [x^n] A(x,t)^n = [1, (2*t + 3), (3*t + 4)*(3*t + 5)/2!, (4*t + 5)*(4*t + 6)*(4*t + 7)/3!, ...]. This sequence has the following specializations:
t = 0: [1, 3, 10, 35, 126, ...] = A001700 (with different offset).
t = 1: [1, 5, 28, 165, 1001, ...] = A025174.
t = 2: [1, 7, 55, 455, 3876, ...] = A224274.
t = 3: [1, 9, 91, 969, 10626, ...] = A163456.
EXAMPLE
Triangle begins
...1
...2 2
...6 15 9
..24 104 144 64
.120 770 1775 1750 625
.720 6264 20880 33480 25920 7776
5040 56196 250096 571095 708295 453789 117649
...
MAPLE
seq(seq(coeff(mul(n*t + k, k = 2 .. n), t, i), i = 0..n-1), n = 1..10);
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Nov 16 2015
STATUS
approved