OFFSET
1,2
COMMENTS
This is a companion triangle to A186695.
Let f(x) = (exp(2*x) + 1)^(-1/2); then the n-th derivative of f equals Sum_{k=1..n} (-1)^k*T(n,k)*(f(x))^(2*k+1). - Groux Roland, May 17 2011
Triangle T(n,k), 1 <= k <= n, given by (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, ...) DELTA (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2013
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.
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
EXAMPLE
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.
MAPLE
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Mar 27 2011
STATUS
approved