This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A303986 #27 May 20 2018 18:33:22
%S A303986 1,1,-2,1,-6,12,1,-12,60,-120,1,-20,180,-840,1680,1,-30,420,-3360,
%T A303986 15120,-30240,1,-42,840,-10080,75600,-332640,665280,1,-56,1512,-25200,
%U A303986 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
%N A303986 Triangle of derivatives of the Niven polynomials evaluated at 0.
%C A303986 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.
%C A303986 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
%D A303986 Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 116-125.
%D A303986 Ivan Niven, Irrational Numbers, Math. Assoc. Am., John Wiley and Sons, New York, 2nd printing 1963, pp. 19-21.
%H A303986 Muniru A Asiru, <a href="/A303986/b303986.txt">Rows n = 0..50</a>
%F A303986 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.
%F A303986 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.
%F A303986 T(n, k) = (-1)^k*binomial(n, n-k)*((n+k)!/n!).
%F A303986 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
%F A303986 T(n, k) = (-1)^k*Pochhammer(n+1, k)*binomial(n, k). - _Peter Luschny_, May 11 2018
%F A303986 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:
%F A303986 T(n, k) = 0 for n < k, T(n, -1) = 0, T(0, 0) = 1 = T(1, 0) and otherwise
%F A303986 T(n, k) = -2*(2*n-1)*T(n-1, k-1) + T(n-2, k). - _Wolfdieter Lang_, May 12 2018
%e A303986 The triangle T(n, k) begins:
%e A303986 n\k 0 1 2 3 4 5 6 7 8 ...
%e A303986 0: 1
%e A303986 1: 1 -2
%e A303986 2: 1 -6 12
%e A303986 3: 1 -12 60 -120
%e A303986 4: 1 -20 180 -840 1680
%e A303986 5: 1 -30 420 -3360 15120 -30240
%e A303986 6: 1 -42 840 -10080 75600 -332640 66528
%e A303986 7: 1 -56 1512 -25200 277200 -1995840 8648640 -17297280
%e A303986 8: 1 -72 2520 -55440 831600 -8648640 60540480 -259459200 518918400
%e A303986 ...
%p A303986 T := (n, k) -> (-1)^k*pochhammer(n+1, k)*binomial(n, k):
%p A303986 seq(print(seq(T(n, k), k=0..n)), n=0..9); # _Peter Luschny_, May 11 2018
%o A303986 (PARI) T(n,k)=(-1)^k*binomial(n,n-k)*binomial(n+k,n)*k! \\ _M. F. Hasler_, May 09 2018
%o A303986 (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
%Y A303986 Row sums are A002119.
%Y A303986 Cf. A098361, A001498, A113025, A113216.
%K A303986 sign,tabl,easy
%O A303986 0,3
%A A303986 _Wolfdieter Lang_, May 07 2018