OFFSET
0,5
COMMENTS
Rows are polynomials in n whose coefficients are in A099599.
From Peter Bala, Aug 19 2013: (Start)
The k-th superdiagonal sequence of this square array occurs as the sequence of numerators in the convergents to a certain continued fraction representation of the constant BesselI(k,2), where BesselI(k,x) is a modified Bessel function of the first kind:
Let d_k(n) = T(n,n+k) = n!*(n+k)!*sum {i = 0..n} 1/(i!*(i+k)!) denote the sequence of entries on the k-th superdiagonal. It satisfies the first-order recurrence equation d_k(n) = n*(n+k)*d_k(n-1) + 1 with d_k(0) = 1 and also the second-order recurrence d_k(n) = (n*(n+k)+1)*d_k(n-1) - (n-1)(n-1+k)*d_k(n-2) with initial conditions d_k(0) = 1 and d_k(1) = k+2. This latter recurrence is also satisfied by the sequence n!*(n+k)!. From this observation we obtain the finite continued fraction expansion d_k(n) = n!*(n+k)!*(1/(k! - k!/((k+2) -(k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) ))))).
Taking the limit as n - > infinity produces a continued fraction representation for the modified Bessel function value BesselI(k,2) = sum {i = 0..inf} 1/(i!*(i+k)!) = 1/(k! - k!/((k+2) -(k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See A070910 for the case k = 0 and A096789 for the case k = 1. (End)
LINKS
E. W. Weisstein, Modified Bessel Function of the First Kind
FORMULA
T(n,k) = sum {i=0..min(n,k)} C(n,i)*C(k,i)*i!^2. The LDU factorization of this square array is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!^2, 1!^2, 2!^2, ... ). Compare with A088699. - Peter Bala, Nov 06 2007
Recurrence equation: T(n,k) = n*k*T(n-1,k-1) + 1 with boundary conditions T(n,0) = T(0,n ) = 1.
Main subdiagonal and main superdiagonal [1, 3, 19, 229, ...] is A228229. - Peter Bala, Aug 19 2013
EXAMPLE
1, 1, 1, 1, 1, 1,
1, 2, 3, 4, 5, 6,
1, 3, 9, 19, 33, 51,
1, 4, 19, 82, 229, 496,
1, 5, 33, 229, 1313, 4581,
1, 6, 51, 496, 4581, 32826,
MAPLE
T := proc(n, k) option remember;
if n = 0 then 1 elif k = 0 then 1
else n*k*thisproc(n-1, k-1) + 1
fi
end:
# Diplay entries by antidiagonals
seq(seq(T(n-k, k), k = 0..n), n = 0..10);
# Peter Bala, Aug 19 2013
MATHEMATICA
T[_, 0] = T[0, _] = 1;
T[n_, k_] := T[n, k] = n k T[n - 1, k - 1] + 1;
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019 *)
CROSSREFS
KEYWORD
AUTHOR
Ralf Stephan, Oct 28 2004
STATUS
approved