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A331430 Triangle read by rows: T(n, k) = (-1)^(k+1)*binomial(n,k)*binomial(n+k,k) (n >= k >= 0).

%I A331430 #106 Oct 18 2022 19:13:32
%S A331430 -1,-1,2,-1,6,-6,-1,12,-30,20,-1,20,-90,140,-70,-1,30,-210,560,-630,
%T A331430 252,-1,42,-420,1680,-3150,2772,-924,-1,56,-756,4200,-11550,16632,
%U A331430 -12012,3432,-1,72,-1260,9240,-34650,72072,-84084,51480,-12870,-1,90,-1980,18480,-90090,252252,-420420,411840,-218790,48620,-1,110,-2970,34320,-210210,756756,-1681680,2333760,-1969110,923780,-184756
%N A331430 Triangle read by rows: T(n, k) = (-1)^(k+1)*binomial(n,k)*binomial(n+k,k) (n >= k >= 0).
%C A331430 This is Table I of Ser (1933), page 92.
%C A331430 From _Petros Hadjicostas_, Jul 09 2020: (Start)
%C A331430 Essentially Ser (1933) in his book (and in particular for Tables I-IV) finds triangular arrays that allow him to express the coefficients of various kinds of series in terms of the coefficients of other series.
%C A331430 He uses Newton's series (or some variation of it), factorial series, and inverse factorial series. Unfortunately, he uses unusual notation, and as a result it is difficult to understand what he is actually doing.
%C A331430 Rivoal (2008, 2009) essentially uses factorial series and transformations to other kinds of series to provide new proofs of the irrationality of log(2), zeta(2), and zeta(3). As a result, the triangular array T(n,k) appears in various parts of his papers.
%C A331430 We believe Table I (p. 92) in Ser (1933), regarding the numbers T(n,k), corresponds to four different formulas. We have deciphered the first two of them. (End)
%D A331430 J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, pp. 92-93.
%H A331430 G. C. Greubel, <a href="/A331430/b331430.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A331430 A. Buhl, <a href="https://www.e-periodica.ch/digbib/view?pid=ens-001:1933:32::554#554">Book review: J. Ser - Les calculs formels des séries de factorielles</a>, L'Enseignement Mathématique, 32 (1933), p. 275.
%H A331430 L. A. MacColl, <a href="https://projecteuclid.org/euclid.bams/1183498080">Review: J. Ser, Les calculs formels des séries de factorielles</a>, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.
%H A331430 L. M. Milne-Thomson, <a href="https://www.jstor.org/stable/3605643">Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars)</a>, The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.
%H A331430 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H A331430 J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%H A331430 Tanguy Rivoal, <a href="http://rivoal.perso.math.cnrs.fr/articles/lagrange.pdf">Applications arithmétiques de l'interpolation lagrangienne</a>, preprint (2008); see pp. 1 and 15.
%H A331430 Tanguy Rivoal, <a href="https://doi.org/10.1142/S1793042109001992">Applications arithmétiques de l'interpolation lagrangienne</a>, Int. J. Number Theory 5.2 (2009), 185-208; see pp. 185 and 199.
%F A331430 T(n,k) can also be written as (-1)^(k+1)*(n+k)!/(k!*k!*(n-k)!).
%F A331430 From _Petros Hadjicostas_, Jul 09 2020: (Start)
%F A331430 Ser's first formula from his Table I (p. 92) is the following:
%F A331430 Sum_{k=0..n} T(n,k)*k!/(x*(x+1)*...*(x+k)) = -(x-1)*(x-2)*...*(x-n)/(x*(x+1)*...*(x+n)).
%F A331430 As a result, Sum_{k=0..n} T(n,k)/binomial(m+k, k) = 0 for m = 1..n.
%F A331430 Ser's second formula from his Table I appears also in Rivoal (2008, 2009) in a slightly different form:
%F A331430 Sum_{k=0..n} T(n,k)/(x + k) = (-1)^(n+1)*(x-1)*(x-2)*...*(x-n)/(x*(x+1)*...*(x+n)).
%F A331430 As a result, for m = 1..n, Sum_{k=0..n} T(n,k)/(m + k) = 0. (End)
%F A331430 T(n,k) = (-1)^(k+1)*FallingFactorial(n+k,2*k)/(k!)^2. - _Peter Luschny_, Jul 09 2020
%F A331430 From _Petros Hadjicostas_, Jul 10 2020: (Start)
%F A331430 _Peter Luschny_'s formula above is essentially the way the numbers T(n,k) appear in Eq. (7) on p. 86 of Ser's (1933) book. Eq. (7) is essentially equivalent to the first formula above (related to Table I on p. 92).
%F A331430 By inverting that formula (in some way), he gets
%F A331430 n!/(x*(x+1)*...*(x+n)) = Sum_{p=0..n} (-1)^p*(2*p+1)*f_p(n+1)*f_p(x), where f_p(x) = (x-1)*...*(x-p)/(x*(x+1)*...*(x+p)). This is equivalent to Eq. (8) on p. 86 of Ser's book.
%F A331430 The rational coefficients A(n,p) = (2*p+1)*f_p(n+1) = (2*p+1)*(n*(n-1)*...*(n+1-p))/((n+1)*...*(n+1+p)) appear in Table II on p. 92 of Ser's book.
%F A331430 If we consider the coefficients T(n,k) and (-1)^(p+1)*A(n,p) as infinite lower triangular matrices, then they are inverses of one another (see the example below). This means that, for m >= s,
%F A331430 Sum_{k=s..m} T(m,k)*(-1)^(s+1)*A(k,s) = I(s=m) = Sum_{k=s..m} (-1)^(k+1)*A(m,k)*T(k,s), where I(s=m) = 1, if s = m, and = 0, otherwise.
%F A331430 Without the (-1)^p, we get the formula
%F A331430 1/(x+n) = Sum_{p=0..n} (2*p+1)*f_p(n+1)*f_p(x),
%F A331430 which apparently is the inversion of the second of Ser's formulas (related to Table I on p. 92).
%F A331430 In all of the above formulas, an empty product is by definition 1, so f_0(x) = 1/x. (End)
%e A331430 Triangle T(n,k) (with rows n >= 0 and columns k=0..n) begins:
%e A331430   -1;
%e A331430   -1,  2;
%e A331430   -1,  6,   -6;
%e A331430   -1, 12,  -30,   20;
%e A331430   -1, 20,  -90,  140,    -70;
%e A331430   -1, 30, -210,  560,   -630,   252;
%e A331430   -1, 42, -420, 1680,  -3150,  2772,   -924;
%e A331430   -1, 56, -756, 4200, -11550, 16632, -12012, 3432;
%e A331430   ...
%e A331430 From _Petros Hadjicostas_, Jul 11 2020: (Start)
%e A331430 Its inverse (from Table II, p. 92) is
%e A331430   -1;
%e A331430   -1/2, 1/2;
%e A331430   -1/3, 1/2,   -1/6;
%e A331430   -1/4, 9/20,  -1/4,  1/20;
%e A331430   -1/5, 2/5,   -2/7,  1/10, -1/70;
%e A331430   -1/6, 5/14, -25/84, 5/36, -1/28,  1/252;
%e A331430   -1/7, 9/28, -25/84, 1/6,  -9/154,  1/84, -1/924;
%e A331430    ... (End)
%t A331430 Table[CoefficientList[-Hypergeometric2F1[-n, n + 1, 1, x], x], {n, 0, 9}] // Flatten (* _Georg Fischer_, Jan 18 2020 after Peter Luschny in A063007 *)
%o A331430 (Magma) /* As triangle: */ [[(-1)^(k+1) * Factorial(n+k) / (Factorial(k) * Factorial(k) * Factorial(n-k)): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Jan 19 2020
%o A331430 (SageMath)
%o A331430 def T(n,k): return (-1)^(k+1)*falling_factorial(n+k,2*k)/factorial(k)^2
%o A331430 flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # _Peter Luschny_, Jul 09 2020
%Y A331430 A063007 is the same triangle without the minus signs, and has much more information.
%Y A331430 Columns 1 and 2 are A002378 and A033487; the last three diagonals are A002544, A002457, A000984.
%Y A331430 Cf. A331431, A331432.
%K A331430 sign,tabl
%O A331430 0,3
%A A331430 _N. J. A. Sloane_, Jan 17 2020
%E A331430 Thanks to _Bob Selcoe_, who noticed a typo in one of the entries, which, when corrected, led to an explicit formula for the whole of Ser's Table I.