A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.

1, 1, -2, 1, -6, 12, 1, -12, 60, -120, 1, -20, 180, -840, 1680, 1, -30, 420, -3360, 15120, -30240, 1, -42, 840, -10080, 75600, -332640, 665280, 1, -56, 1512, -25200, 277200, -1995840, 8648640, -17297280, 1, -72, 2520, -55440, 831600, -8648640, 60540480, -259459200, 518918400, 1, -90, 3960, -110880, 2162160, -30270240, 302702400, -2075673600, 8821612800, -17643225600, 1, -110, 5940, -205920, 5045040, -90810720, 1210809600, -11762150400, 79394515200, -335221286400, 670442572800

Offset: 0

Author

Wolfdieter Lang, May 07 2018

Keywords

• sign
• tabl
• easy

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

Examples

			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
...
		

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

Crossrefs

Row sums are A002119.

Cf. A098361, A001498, A113025, A113216.

Programs

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

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

PARI

T(n,k)=(-1)^k*binomial(n,n-k)*binomial(n+k,n)*k! \\ M. F. Hasler, May 09 2018

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