OFFSET
0,3
COMMENTS
The Niven potentials N(n, x) = (1/n!)*x^n*(1 - x)^n = Sum_{k=0..n} (-1)^k * x^(n+k)/((n-k)!*k!), with (n-k)!*k! = A098361(n, k), are used to prove the irrationality of Pi^2 (hence Pi). See the Niven and Havil references.
The row polynomials R(n, x) = Sum_{k=0..n} T(n, k) *x^k are R(n, x) = y_n(-2*x), with the Bessel polynomials of Krall and Frink y_n(x) with coefficients given in A001498. There the references are given. - Wolfdieter Lang, May 12 2018
REFERENCES
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.
LINKS
Muniru A Asiru, Rows n = 0..50
FORMULA
T(n, k) = (d/dx)^(n+k) N(n, x) |_{x=0} =: N^{(n+k)}(n, 0), with N(n, x) = (1/n!)*x^n*(1 - x)^n, for n >= 0, k = 0..n.
N^{(n+k)}(n, 1) = (-1)^(n+k)*T(n, k), which has for even n the unsigned rows, and for odd n the unsigned row entries with negative signs.
T(n, k) = (-1)^k*binomial(n, n-k)*((n+k)!/n!).
T(n, k) = (-1)^k*A113025(n,k) with A113025(n,k) = (n+k)!/(k!*(n-k)!) = abs(A113216(n,k)). - M. F. Hasler, May 09 2018
T(n, k) = (-1)^k*Pochhammer(n+1, k)*binomial(n, k). - Peter Luschny, May 11 2018
Recurrence: from the one of the row polynomials R(n, x) = y_n(-2*x): R(n, x) = -2*(2*n-1)*x*R(n-1, x) + R(n-2, x), with R(-1, x) = 1 = R(0, x) = 1, n >= 1 (see A001498), this becomes, for n >= 0, k = 0..n:
T(n, k) = 0 for n < k, T(n, -1) = 0, T(0, 0) = 1 = T(1, 0) and otherwise
T(n, k) = -2*(2*n-1)*T(n-1, k-1) + T(n-2, k). - Wolfdieter Lang, May 12 2018
EXAMPLE
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 ...
0: 1
1: 1 -2
2: 1 -6 12
3: 1 -12 60 -120
4: 1 -20 180 -840 1680
5: 1 -30 420 -3360 15120 -30240
6: 1 -42 840 -10080 75600 -332640 66528
7: 1 -56 1512 -25200 277200 -1995840 8648640 -17297280
8: 1 -72 2520 -55440 831600 -8648640 60540480 -259459200 518918400
...
MAPLE
T := (n, k) -> (-1)^k*pochhammer(n+1, k)*binomial(n, k):
seq(print(seq(T(n, k), k=0..n)), n=0..9); # Peter Luschny, May 11 2018
PROG
(PARI) T(n, k)=(-1)^k*binomial(n, n-k)*binomial(n+k, n)*k! \\ M. F. Hasler, May 09 2018
(GAP) Flat(List([0..10], n->List([0..n], k->(-1)^k*Binomial(n, n-k)*Factorial(n+k)/Factorial(n)))); # Muniru A Asiru, May 15 2018
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 07 2018
STATUS
approved