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%I #79 Oct 18 2022 19:12:14
%S 1,-2,6,3,-24,30,-4,60,-180,140,5,-120,630,-1120,630,-6,210,-1680,
%T 5040,-6300,2772,7,-336,3780,-16800,34650,-33264,12012,-8,504,-7560,
%U 46200,-138600,216216,-168168,51480,9,-720,13860,-110880,450450,-1009008,1261260,-823680,218790
%N Triangle read by rows: T(n,k) = (-1)^(n+k)*(n+k+1)*binomial(n,k)*binomial(n+k,k) for n >= k >= 0.
%C Tables I, III, IV on pages 92 and 93 of Ser have integer entries and are A331430, A331431 (the present sequence), and A331432.
%C Given the system of equations 1 = Sum_{j=0..n} H(i, j) * x(j) for i = 2..n+2 where H(i,j) = 1/(i+j-1) for 1 <= i,j <= n is the n X n Hilbert matrix, then the solutions are x(j) = T(n, j). - _Michael Somos_, Mar 20 2020 [Corrected by _Petros Hadjicostas_, Jul 09 2020]
%D J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93. See Table III.
%H G. C. Greubel, <a href="/A331431/b331431.txt">Rows n = 0..50 of the triangle, flattened</a>
%H 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 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 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 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages.)
%F T(n, 0) = (-1)^n*A000027(n+1).
%F T(n, 1) = A331433(n-1) = (-1)^(n+1)*A007531(n+2).
%F T(n, 2) = A331434(n-2) = (-1)^n*A054559(n+3).
%F T(n, n-2) = A002738(n-2).
%F T(n, n-1) = (-1)*A002736(n).
%F T(n, n) = A002457(n).
%F T(2*n, n) = (-1)^n*(3*n+1)!/(n!)^3 = (-1)^n*A331322(n).
%F Sum_{k=0..n} T(n, k) = A000290(n+1) (row sums).
%F Sum_{k=0..n}((-1)^k*T(n, k) = (-1)^n*A108666(n+1) (alternating row sums).
%F Sum_{k=0..n} T(n-k, k) = (-1)^n*A109188(n+1) (diagonal sums).
%F 2^n*Sum_{k=0..n} T(n, k)/2^k = (-1)^floor(n/2)*A100071(n+1) (positive half sums).
%F (-2)^n*Sum_{k=0..n} T(n, k)/(-2)^k = A331323(n) (negative half sums).
%F T(n, k) = ((2*k+1)!/(k!)^2)*[x^(n-k)] (1+x)^(-2*(k+1)). - _Georg Fischer_ and _Peter Luschny_, Jan 18 2020
%F T(n,k) = (-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!), for n >= k >= 0. - _N. J. A. Sloane_, Jan 18 2020
%F From _Petros Hadjicostas_, Jul 09 2020: (Start)
%F _Michael Somos_'s formulas above can be restated as
%F Sum_{k=0..n} T(n,k)/(i+k) = 1 for i = 1..n+1.
%F These are special cases of the following formula that is alluded to (in some way) in Ser's book:
%F 1 - Sum_{k=0..n} T(n,k)/(x + k) = (x-1)*...*(x-(n + 1))/(x*(x+1)*...*(x+n)).
%F Because T(n,k) = (-1)^(n+1)*(n + k + 1)*A331430(n,k) and Sum_{k=0..n} A331430(n,k) = (-1)^(n+1), one may derive this formula from Ser's second formula stated in A331430. (End)
%F T(2*n+1, n) = (-2)*(-27)^n*Pochhammer(4/3, n)*Pochhammer(5/3, n)/(n!*(n+1)!). - _G. C. Greubel_, Mar 22 2022
%e Triangle begins:
%e 1;
%e -2, 6;
%e 3, -24, 30;
%e -4, 60, -180, 140;
%e 5, -120, 630, -1120, 630;
%e -6, 210, -1680, 5040, -6300, 2772;
%e 7, -336, 3780, -16800, 34650, -33264, 12012;
%e -8, 504, -7560, 46200, -138600, 216216, -168168, 51480;
%e 9, -720, 13860, -110880, 450450, -1009008, 1261260, -823680, 218790;
%e ...
%p gf := k -> (1+x)^(-2*(k+1)): ser := k -> series(gf(k), x, 32):
%p T := (n, k) -> ((2*k+1)!/(k!)^2)*coeff(ser(k), x, n-k):
%p seq(seq(T(n,k), k=0..n),n=0..7); # _Peter Luschny_, Jan 18 2020
%p S:=(n,k)->(-1)^(n+k)*(n+k+1)!/((k!)^2*(n-k)!);
%p rho:=n->[seq(S(n,k),k=0..n)];
%p for n from 0 to 14 do lprint(rho(n)); od: # _N. J. A. Sloane_, Jan 18 2020
%t Table[(-1)^(n+k)*(n+k+1)*Binomial[2*k,k]*Binomial[n+k,n-k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 22 2022 *)
%o (Magma) [(-1)^(n+k)*(k+1)*(2*k+1)*Binomial(n+k+1,n-k)*Catalan(k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Mar 22 2022
%o (Sage) flatten([[(-1)^(n+k)*(2*k+1)*binomial(2*k,k)*binomial(n+k+1,n-k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Mar 22 2022
%Y Columns 1 is A331433 or equally A007531, column 2 is A331434 or equally A054559; the last three diagonals are A002738, A002736, A002457.
%Y Cf. A000290 (row sums), A002457,, A100071, A108666 (alternating row sums), A109188 (diagonal sums), A331322, A331323, A331430, A331432.
%K sign,tabl
%O 0,2
%A _N. J. A. Sloane_, Jan 17 2020
%E Several typos in the data corrected by _Georg Fischer_ and _Peter Luschny_, Jan 18 2020
%E Definition changed by _N. J. A. Sloane_, Jan 18 2020
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