A187075 A Galton triangle: T(n,k) = 2*k*T(n-1,k) + (2*k-1)*T(n-1,k-1).
1, 2, 3, 4, 18, 15, 8, 84, 180, 105, 16, 360, 1500, 2100, 945, 32, 1488, 10800, 27300, 28350, 10395, 64, 6048, 72240, 294000, 529200, 436590, 135135, 128, 24384, 463680, 2857680, 7938000, 11060280, 7567560, 2027025, 256, 97920, 2904000, 26107200, 105099120, 220041360, 249729480, 145945800, 34459425
Offset: 1
Examples
Triangle begins n\k.|...1.....2......3......4......5......6 =========================================== ..1.|...1 ..2.|...2.....3 ..3.|...4....18.....15 ..4.|...8....84....180....105 ..5.|..16...360...1500...2100....945 ..6.|..32..1488..10800..27300..28350..10395 .. Examples of recurrence relation: T(4,3) = 6*T(3,3) + 5*T(3,2) = 6*15 + 5*18 = 180; T(6,4) = 8*T(5,4) + 7*T(5,3) = 8*2100 + 7*1500 = 27300.
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows
- Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, arXiv:1204.4963v3 [math.CO]
- Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
- E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.
Programs
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Maple
A187075 := proc(n, k) option remember; if k < 1 or k > n then 0; elif k = 1 then 2^(n-1); else 2*k*procname(n-1, k) + (2*k-1)*procname(n-1, k-1) ; end if; end proc:seq(seq(A187075(n,k),k = 1..n),n = 1..10);
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Mathematica
Flatten[Table[2^(n - 2*k)*Binomial[2 k, k]*k!*StirlingS2[n, k], {n, 10}, {k, 1, n}]] (* G. C. Greubel, Jun 17 2016 *) -
Sage
# uses[delehamdelta from A084938] # Adds a first column (1,0,0,0, ...). def A187075_triangle(n): return delehamdelta([(i+1)*int(is_even(i+1)) for i in (0..n)], [i+1 for i in (0..n)]) A187075_triangle(4) # Peter Luschny, Oct 20 2013
Formula
T(n,k) = 2^(n-2*k)*binomial(2k,k)*k!*Stirling2(n,k).
Recurrence relation T(n,k) = 2*k*T(n-1,k) + (2*k-1)*T(n-1,k-1) with boundary conditions T(1,1) = 1, T(1,k) = 0 for k >= 2.
G.f.: F(x,t) = 1/sqrt((1+x)-x*exp(2*t)) - 1 = Sum_{n >= 1} R(n,x)*t^n/n! = x*t + (2*x+3*x^2)*t^2/2! + (4*x+18*x^2+15*x^3)*t^3/3! + ....
The g.f. F(x,t) satisfies the partial differential equation dF/dt = 2*(x+x^2)*dF/dx + x*F.
The row polynomials R(n,x) satisfy the recursion R(n+1,x) = 2*(x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x.
O.g.f. for column k: (2k-1)!!*x^k/Product_{m = 1..k} (1-2*m*x) (compare with A075497). T(n,k) = (2*k-1)!!*A075497(n,k).
The row polynomials R(n,x) = Sum_{k = 1..n} T(n,k)*x^k satisfy R(n,-x-1) = (-1)^n*(1+x)/x*P(n,x) where P(n,x) is the n-th row polynomial of A186695. We also have R(n,x/(1-x)) = (x/(1-x)^n)*Q(n-1,x) where Q(n,x) is the n-th row polynomial of A156919.
T(n,k) = 2^(n-k)*A211608(n,k). - Philippe Deléham, Oct 20 2013
Comments