login
The OEIS is supported by the many generous donors 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

%I #42 Mar 14 2024 11:08:24

%S 1,1,1,1,5,1,1,21,12,1,1,85,105,22,1,1,341,820,325,35,1,1,1365,6081,

%T 4070,780,51,1,1,5461,43932,46781,14210,1596,70,1,1,21845,312985,

%U 511742,231511,39746,2926,92,1,1,87381,2212740,5430405,3521385,867447,95340,4950,117,1

%N Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.

%C 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.

%C Riordan array [exp(x), (exp(3x)-1)/3]. - _Paul Barry_, Nov 26 2008

%C From _Peter Bala_, Jan 27 2015: (Start)

%C 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.

%C 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.

%C 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)

%C Named after the English scientist Francis Galton (1822-1911). - _Amiram Eldar_, Jun 13 2021

%H Peter Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>, 2015.

%H Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See p. 8.

%H Ruedi Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two Analogues of a Classical Sequence</a>, Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.8. [_Paul Barry_, Nov 26 2008]

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

%F E.g.f.: exp(x)*exp((y/3)*(exp(3x)-1)). - _Paul Barry_, Nov 26 2008

%F 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

%F 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

%F From _Peter Bala_, Jan 27 2015: (Start)

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

%F 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!.

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

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

%e The triangle starts in row n=1 as:

%e 1;

%e 1,1;

%e 1,5,1;

%e 1,21,12,1;

%e 1,85,105,22,1;

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

%e x^3 = 1 + 21*(x - 1) + 12*(x - 1)*(x - 4) + (x - 1)*(x - 4)*(x - 7). - _Peter Bala_, Jan 27 2015

%p 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:

%p seq( seq(A111577(n,k),k=1..n), n=1..10) ; # _R. J. Mathar_, Aug 22 2009

%t T[_, 1] = 1; T[n_, n_] = 1;

%t T[n_, k_] := T[n, k] = T[n-1, k-1] + (3k-2) T[n-1, k];

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* _Jean-François Alcover_, Jun 13 2019 *)

%Y Cf. A008277, A039755, A075498, A225468.

%Y Cf. A282629, A284861, A225117.

%K nonn,easy,tabl,changed

%O 1,5

%A _Gary W. Adamson_, Aug 07 2005

%E Edited and extended by _R. J. Mathar_, Aug 22 2009

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)