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 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 R. Suter, Two Analogues of a Classical Sequence, Journal of Integer Sequences, Article 00.1.8. [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 MATHEMATICA T[_, 1] = 1; T[n_, n_] = 1; T[n_, k_] := T[n, k] = T[n-1, k-1] + (3k-2) T[n-1, k]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* Jean-François Alcover, Jun 13 2019 *) 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

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Last modified November 21 04:33 EST 2019. Contains 329350 sequences. (Running on oeis4.)