This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A111577 #57 Aug 03 2025 05:00:54
%S A111577 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 A111577 4070,780,51,1,1,5461,43932,46781,14210,1596,70,1,1,21845,312985,
%U A111577 511742,231511,39746,2926,92,1,1,87381,2212740,5430405,3521385,867447,95340,4950,117,1
%N A111577 Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.
%C A111577 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 A111577 Riordan array [exp(x), (exp(3x)-1)/3]. - _Paul Barry_, Nov 26 2008
%C A111577 From _Peter Bala_, Jan 27 2015: (Start)
%C A111577 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 A111577 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 A111577 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 A111577 Named after the English scientist Francis Galton (1822-1911). - _Amiram Eldar_, Jun 13 2021
%C A111577 This is the array of (r, β)-Stirling numbers for r = 1 and β = 3. See Corcino. - _Peter Bala_, Feb 26 2025
%H A111577 Peter Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>, 2015.
%H A111577 Peter Bala, <a href="/A143494/a143494.pdf">Factorising (r,b)-Stirling arrays</a>
%H A111577 Roberto B. Corcino, <a href="https://www.researchgate.net/publication/354943509">The (r, β)-Stirling Numbers</a>, The Mindanao Forum, Vol. XIV, No.2, pp. 91-99, 1999.
%H A111577 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 A111577 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 A111577 T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k).
%F A111577 E.g.f.: exp(x)*exp((y/3)*(exp(3x)-1)). - _Paul Barry_, Nov 26 2008
%F A111577 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 A111577 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 A111577 From _Peter Bala_, Jan 27 2015: (Start)
%F A111577 T(n,k) = Sum_{i = 0..n-1} 3^(i-k+1)*binomial(n-1,i)*Stirling2(i,k-1).
%F A111577 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 A111577 O.g.f. column k (with offset 0): 1/( (1 - x)*(1 - 4*x)*...*(1 - (3*k + 1)*x) ). (End)
%e A111577 T(5,3) = T(4,2) + 7*T(4,3) = 21 + 7*12 = 105.
%e A111577 The triangle starts in row n = 1 as:
%e A111577 1;
%e A111577 1, 1;
%e A111577 1, 5, 1;
%e A111577 1, 21, 12, 1;
%e A111577 1, 85, 105, 22, 1;
%e A111577 Connection constants: Row 4: [1, 21, 12, 1] so
%e A111577 x^3 = 1 + 21*(x - 1) + 12*(x - 1)*(x - 4) + (x - 1)*(x - 4)*(x - 7). - _Peter Bala_, Jan 27 2015
%e A111577 From _Peter Bala_, Feb 26 2025: (Start)
%e A111577 The array factorizes as
%e A111577 /1 \ /1 \/1 \/1 \
%e A111577 |1 1 | |1 1 ||0 1 ||0 1 |
%e A111577 |1 5 1 | = |1 4 1 ||0 1 1 ||0 0 1 | ...
%e A111577 |1 21 12 1 | |1 13 7 1 ||0 1 4 1 ||0 0 1 1 |
%e A111577 |1 85 105 22 1| |1 44 34 10 1||0 1 13 7 1 ||0 0 1 4 1 |
%e A111577 |... | |... ||... ||... |
%e A111577 where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - x), x/(1 - 3*x)). Cf. A193843. (End)
%p A111577 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 A111577 seq( seq(A111577(n,k),k=1..n), n=1..10) ; # _R. J. Mathar_, Aug 22 2009
%t A111577 T[_, 1] = 1; T[n_, n_] = 1;
%t A111577 T[n_, k_] := T[n, k] = T[n-1, k-1] + (3k-2) T[n-1, k];
%t A111577 Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* _Jean-François Alcover_, Jun 13 2019 *)
%Y A111577 Cf. A008277, A039755, A075498, A225468.
%Y A111577 Cf. A282629, A284861, A225117.
%K A111577 nonn,easy,tabl
%O A111577 1,5
%A A111577 _Gary W. Adamson_, Aug 07 2005
%E A111577 Edited and extended by _R. J. Mathar_, Aug 22 2009