1,5

In triangles of analogs to Stirling numbers of the second kind, the multipliers of T(n-1,k) in the recurrence are terms in arithmetic sequences: in Pascal's triangle A007318, the multiplier = 1. In triangle A008277, the Stirling numbers of the second kind, the multipliers are in the set (1,2,3...). For this sequence here, the multipliers are from A016777.

Riordan array [exp(x), (exp(3x)-1)/3]. - Paul Barry, Nov 26 2008

From Peter Bala, Jan 27 2015: (Start)

Working with an offset of 0, this is the triangle of connection constants between the polynomial basis sequences {x^n}n>=0 and { n!*3^n*binomial((x - 1)/3,n) }n>=0. An example is given below.

Call this array M and let P denote Pascal's triangle A007318, then P * M = A225468, P^2 * M = A075498. Also P^(-1) * M is a shifted version of A075498.

This triangle is the particular case a = 3, b = 0, c = 1 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. (End)

Table of n, a(n) for n=1..55.

P. Bala, A 3 parameter family of generalized Stirling numbers

R. Suter, Two Analogues of a Classical Sequence, Journal of Integer Sequences, Article 00.1.8 [From Paul Barry, Nov 26 2008]

T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k).

E.g.f.: exp(x)*exp((y/3)*(exp(3x)-1)). - Paul Barry, Nov 26 2008

Let f(x) = exp(1/3*exp(3*x)+x). Then, with an offset of 0, the row polynomials R(n,x) are given by R(n,exp(3*x)) = 1/f(x)*(d/dx)^n(f(x)). Similar formulas hold for A008277, A039755, A105794, A143494 and A154537. - Peter Bala, Mar 01 2012

T(n, k) = 1/(3^k*k!)*sum_{j=0..k}((-1)^(k-j)*binomial(k,j)*(3*j+1)^n). - Peter Luschny, May 20 2013

From Peter Bala, Jan 27 2015: (Start)

T(n,k) = sum {i = 0..n-1} 3^(i-k+1)*binomial(n-1,i)*Stirling2(i,k-1).

O.g.f. for n-th diagonal: exp(-x/3)*sum {k >= 0} (3*k + 1)^(k + n - 1)*((x/3*exp(-x))^k)/k!.

O.g.f. column k (with offset 0): 1/( (1 - x)*(1 - 4*x)...(1 - (3*k + 1)*x ). (End)

T(5,3) = T(4,2)+7*T(4,3) = 21 + 7*12 = 105.

The triangle starts in row n=1 as:

1;

1,1;

1,5,1;

1,21,12,1;

1,85,105,22,1;

Connection constants: Row 4: [1, 21, 12, 1] so

x^3 = 1 + 21*(x - 1) + 12*(x - 1)*(x - 4) + (x - 1)*(x - 4)*(x - 7). - Peter Bala, Jan 27 2015

A111577 := proc(n, k) option remember; if k = 1 or k = n then 1; else procname(n-1, k-1)+(3*k-2)*procname(n-1, k) ; fi; end:

seq( seq(A111577(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Aug 22 2009

Cf. A008277, A039755, A075498, A225468.

Sequence in context: A144397 A047909 A171243 * A176242 A036969 A080249

Adjacent sequences:  A111574 A111575 A111576 * A111578 A111579 A111580

nonn,easy,tabl

Gary W. Adamson, Aug 07 2005

Edited and extended by R. J. Mathar, Aug 22 2009

approved