login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A111577 Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows. 15
1, 1, 1, 1, 5, 1, 1, 21, 12, 1, 1, 85, 105, 22, 1, 1, 341, 820, 325, 35, 1, 1, 1365, 6081, 4070, 780, 51, 1, 1, 5461, 43932, 46781, 14210, 1596, 70, 1, 1, 21845, 312985, 511742, 231511, 39746, 2926, 92, 1, 1, 87381, 2212740, 5430405, 3521385, 867447, 95340, 4950, 117, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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)

LINKS

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]

FORMULA

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)

EXAMPLE

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

MAPLE

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

CROSSREFS

Cf. A008277, A039755, A075498, A225468.

Cf. A282629, A284861, A225117.

Sequence in context: A144397 A047909 A171243 * A176242 A036969 A080249

Adjacent sequences:  A111574 A111575 A111576 * A111578 A111579 A111580

KEYWORD

nonn,easy,tabl

AUTHOR

Gary W. Adamson, Aug 07 2005

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 13 13:15 EST 2018. Contains 317149 sequences. (Running on oeis4.)