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%I #46 Feb 29 2024 10:48:07
%S 1,-1,1,4,-5,1,-36,49,-14,1,576,-820,273,-30,1,-14400,21076,-7645,
%T 1023,-55,1,518400,-773136,296296,-44473,3003,-91,1,-25401600,
%U 38402064,-15291640,2475473,-191620,7462,-140,1,1625702400,-2483133696,1017067024,-173721912,14739153,-669188,16422,-204,1
%N Triangle read by rows: matrix inverse of the central factorial numbers T(2n, 2k) (A036969).
%C This is a signed version of A008955 with rows in reverse order. - _Peter Luschny_, Feb 04 2012
%H M. W. Coffey, M. C. Lettington, <a href="http://arxiv.org/abs/1510.05402">On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat</a>, arXiv:1510.05402 [math.NT], 2015.
%F T(n, k) = (-1)^(n-k)*A008955(n, n-k). - _Peter Luschny_, Feb 05 2012
%F T(n, k) = Sum_{i=k-n..n-k} (-1)^(n-k+i)*s(n,k+i)*s(n,k-i) = Sum_{i=0..2*k} (-1)^(n+i)*s(n,i)*s(n,2*k-i), where s(n,k) are Stirling numbers of the first kind, A048994. - _Mircea Merca_, Apr 07 2012
%F From _Peter Bala_, Aug 29 2012: (Start)
%F T(n, k) = T(n-1, k-1) - (n-1)^2*T(n-1, k). (Recurrence equation.)
%F Let E(x) = cosh(sqrt(2*x)) = Sum_{n >= 0} x^n/{(2*n)!/2^n} and
%F L(x) = 2*{arcsinh(sqrt(x/2))}^2 = Sum_{n >=1} (-1)^n*(n-1)!^2*x^n/{(2*n)!/2^n}.
%F L(x) is the compositional inverse of E(x) - 1.
%F A generating function for the triangle is E(t*L(x)) = 1 + t*x + t*(-1 + t)*x^2/6 + t*(4 - 5*t + t^2)*x^3/90 + ..., where the sequence of denominators [1,1,6,90,...] is given by (2*n)!/2^n. Cf. A008275 with generating function exp(t*log(1+x)).
%F The e.g.f. is E(t*L(x^2/2)) = cosh(2*sqrt(t)*arcsinh(x/2)) = 1 + t*x^2/2! + t*(t-1)*x^4/4! + t*(t-1)*(t-4)*x^6/6! + .... (End)
%F From _Peter Luschny_, Feb 29 2024: (Start)
%F T(n, k) = [z^(2*k)] z^2*Product_{j=1..n-1} (z^2 - j^2).
%F T(n, k) = (2*n)! * [t^k] [x^(2*n)] (w^sqrt(t) + w^(-sqrt(t)))/2 where w = (x/2 + sqrt(1 + (x/2)^2)^2. (End)
%e Triangle starts:
%e [1] 1;
%e [2] -1, 1;
%e [3] 4, -5, 1;
%e [4] -36, 49, -14, 1;
%e [5] 576, -820, 273, -30, 1;
%e [6] -14400, 21076, -7645, 1023, -55, 1;
%e [7] 518400, -773136, 296296, -44473, 3003, -91, 1;
%e [8] -25401600, 38402064, -15291640, 2475473, -191620, 7462, -140, 1;
%p # From _Peter Luschny_, Feb 29 2024: (Start)
%p ogf := n -> local j; z^2*mul(z^2 - j^2, j = 1..n-1):
%p Trow := n -> local k; seq(coeff(expand(ogf(n)), z, 2*k), k = 1..n):
%p # Alternative:
%p f := w -> (w^sqrt(t) + w^(-sqrt(t)))/2: egf := f((x/2 + sqrt(1 + (x/2)^2))^2):
%p ser := series(egf, x, 20): cx := n -> coeff(ser, x, 2*n):
%p Trow := n -> local k; seq((2*n)!*coeff(cx(n), t, k), k = 1..n): # (End)
%t rows = 10;
%t t[n_, k_] := 2*Sum[j^(2*n)*(-1)^(k - j)/((k - j)!*(k + j)!), {j, 1, k}];
%t T = Table[t[n, k], {n, 1, rows}, {k, 1, rows}] // Inverse;
%t Table[T[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 14 2018 *)
%o (PARI) select(concat(Vec(matrix(10,10,n,k,T(n,k)/*from A036969*/)~^-1)), x->x)
%o (Sage)
%o def A204579(n, k): return (-1)^(n-k)*A008955(n, n-k)
%o for n in (0..7): print([A204579(n, k) for k in (0..n)]) # _Peter Luschny_, Feb 05 2012
%Y Cf. A036969, A008955, A008275, A121408, A001044 (column 1), A101686 (alternating row sums), A234324 (central terms).
%K sign,tabl
%O 1,4
%A _M. F. Hasler_, Feb 03 2012
%E Typo in data corrected by _Peter Luschny_, Feb 05 2012
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