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%I #97 Feb 16 2025 08:32:58
%S 1,1,1,1,3,1,1,7,5,1,1,13,19,7,1,1,21,55,37,9,1,1,31,131,147,61,11,1,
%T 1,43,271,471,309,91,13,1,1,57,505,1281,1251,561,127,15,1,1,73,869,
%U 3067,4251,2751,923,169,17,1,1,91,1405,6637,12559,11253,5321,1415,217,19,1
%N Square array, read by antidiagonals, where row n equals the crystal ball sequence for the A_n lattice.
%C Compare to the corresponding array A108553 of crystal ball sequences for D_n lattice.
%C From _Peter Bala_, Jul 18 2008: (Start)
%C Row reverse of A099608.
%C This array has a remarkable relationship with the constant zeta(2). The row, column and diagonal entries of the array occur in series acceleration formulas for zeta(2).
%C For the entries in row n we have zeta(2) = 2*(1 - 1/2^2 + 1/3^2 - ... + (-1)^(n+1)/n^2) + (-1)^n*Sum_{k >= 1} 1/(k^2*T(n,k-1)*T(n,k)). For example, n = 4 gives zeta(2) = 2*(1 - 1/4 + 1/9 - 1/16) + 1/(1*21) + 1/(4*21*131) + 1/(9*131*471) + ... . See A142995 for further details.
%C For the entries in column k we have zeta(2) = (1 + 1/4 + 1/9 + ... + 1/k^2) + 2*Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n-1,k)*T(n,k)). For example, k = 4 gives zeta(2) = (1 + 1/4 + 1/9 + 1/16) + 2*(1/(1*9) - 1/(4*9*61) + 1/(9*61*309) - ... ). See A142999 for further details.
%C Also, as consequence of Apery's proof of the irrationality of zeta(2), we have a series acceleration formula along the main diagonal of the table: zeta(2) = 5 * Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n,n)*T(n-1,n-1)) = 5*(1/3 - 1/(2^2*3*19) + 1/(3^2*19*147) - ...).
%C There also appear to be series acceleration results along other diagonals. For example, for the main subdiagonal, calculation supports the result zeta(2) = 2 - Sum_{n >= 1} (-1)^(n+1)*(n^2+(2*n+1)^2)/(n^2*(n+1)^2*T(n,n-1)*T(n+1,n)) = 2 - 10/(2^2*7) + 29/(6^2*7*55) - 58/(12^2*55*471) + ..., while for the main superdiagonal we appear to have zeta(2) = 1 + Sum_{n >= 1} (-1)^(n+1)*((n+1)^2 + (2*n+1)^2)/(n^2*(n+1)^2*T(n-1,n)*T(n,n+1)) = 1 + 13/(2^2*5) - 34/(6^2*5*37) + 65/(12^2*37*309) - ... .
%C Similar series acceleration results hold for Apery's constant zeta(3) involving the crystal ball sequences for the product lattices A_n x A_n; see A143007 for further details. Similar results also hold between the constant log(2) and the crystal ball sequences of the hypercubic lattices A_1 x...x A_1 and between log(2) and the crystal ball sequences for lattices of type C_n ; see A008288 and A142992 respectively for further details. (End)
%C This array is the Hilbert transform of triangle A008459 (see A145905 for the definition of the Hilbert transform). - _Peter Bala_, Oct 28 2008
%H Alois P. Heinz, <a href="/A108625/b108625.txt">Antidiagonals n = 0..140, flattened</a>
%H R. Bacher, P. de la Harpe, and B. Venkov, <a href="http://archive.numdam.org/item/AIF_1999__49_3_727_0">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H Joseph T. Iosue, T. C. Mooney, Adam Ehrenberg, and Alexey V. Gorshkov, <a href="https://arxiv.org/abs/2311.13479">Projective toric designs, difference sets, and quantum state designs</a>, arXiv:2311.13479 [quant-ph], 2023. See page 6.
%H Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.
%H A. van der Poorten, <a href="http://pracownicy.uksw.edu.pl/mwolf/Poorten_MI_195_0.pdf"> A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report.</a> Math. Intelligencer 1 (1978/79), no 4, 195-203.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AperyNumber.html">Apéry number</a>.
%F T(n, k) = Sum_{i=0..k} C(n, i)^2 * C(n+k-i, k-i).
%F G.f. for row n: (Sum_{i=0..n} C(n, i)^2 * x^i)/(1-x)^(n+1).
%F Sum_{k=0..n} T(n-k, k) = A108626(n) (antidiagonal sums).
%F From _Peter Bala_, Jul 23 2008 (Start):
%F O.g.f. row n: 1/(1 - x)*Legendre_P(n,(1 + x)/(1 - x)).
%F G.f. for square array: 1/sqrt((1 - x)*((1 - t)^2 - x*(1 + t)^2)) = (1 + x + x^2 + x^3 + ...) + (1 + 3*x + 5*x^2 + 7*x^3 + ...)*t + (1 + 7*x + 19*x^2 + 37*x^3 + ...)*t^2 + ... . Cf. A142977.
%F Main diagonal is A005258.
%F Recurrence relations:
%F Row n entries: (k+1)^2*T(n,k+1) = (2*k^2+2*k+n^2+n+1)*T(n,k) - k^2*T(n,k-1), k = 1,2,3,... ;
%F Column k entries: (n+1)^2*T(n+1,k) = (2*k+1)*(2*n+1)*T(n,k) + n^2*T(n-1,k), n = 1,2,3,... ;
%F Main diagonal entries: (n+1)^2*T(n+1,n+1) = (11*n^2+11*n+3)*T(n,n) + n^2*T(n-1,n-1), n = 1,2,3,... .
%F Series acceleration formulas for zeta(2):
%F Row n: zeta(2) = 2*(1 - 1/2^2 + 1/3^2 - ... + (-1)^(n+1)/n^2) + (-1)^n*Sum_{k >= 1} 1/(k^2*T(n,k-1)*T(n,k));
%F Column k: zeta(2) = 1 + 1/2^2 + 1/3^2 + ... + 1/k^2 + 2*Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n-1,k)*T(n,k));
%F Main diagonal: zeta(2) = 5 * Sum_{n >= 1} (-1)^(n+1)/(n^2*T(n-1,n-1)*T(n,n)).
%F Conjectural result for superdiagonals: zeta(2) = 1 + 1/2^2 + ... + 1/k^2 + Sum_{n >= 1} (-1)^(n+1) * (5*n^2 + 6*k*n + 2*k^2)/(n^2*(n+k)^2*T(n-1,n+k-1)*T(n,n+k)), k = 0,1,2... .
%F Conjectural result for subdiagonals: zeta(2) = 2*(1 - 1/2^2 + ... + (-1)^(k+1)/k^2) + (-1)^k*Sum_{n >= 1} (-1)^(n+1)*(5*n^2 + 4*k*n + k^2)/(n^2*(n+k)^2*T(n+k-1,n-1)*T(n+k,n)), k = 0,1,2... .
%F Conjectural congruences: the main superdiagonal numbers S(n) := T(n,n+1) appear to satisfy the supercongruences S(m*p^r - 1) = S(m*p^(r-1) - 1) (mod p^(3*r)) for all primes p >= 5 and all positive integers m and r. If p is prime of the form 4*n + 1 we can write p = a^2 + b^2 with a an odd number. Then calculation suggests the congruence S((p-1)/2) == 2*a^2 (mod p). (End)
%F From _Michael Somos_, Jun 03 2012: (Start)
%F T(n, k) = hypergeom([-n, -k, n + 1], [1, 1], 1).
%F T(n, n-1) = A208675(n).
%F T(n+1, n) = A108628(n). (End)
%F T(n, k) = binomial(n, k)*hypergeom([-k, k - n, k - n], [1, -n], 1). - _Peter Luschny_, Feb 10 2018
%F From _Peter Bala_, Jun 23 2023: (Start)
%F T(n, k) = Sum_{i = 0..k} (-1)^i * binomial(n, i)*binomial(n+k-i, k-i)^2.
%F T(n, k) = binomial(n+k, k)^2 * hypergeom([-n, -k, -k], [-n - k, -n - k], 1). (End)
%F From _Peter Bala_, Jun 28 2023; (Start)
%F T(n,k) = the coefficient of (x^n)*(y^k)*(z^n) in the expansion of 1/( (1 - x - y)*(1 - z ) - x*y*z ).
%F T(n,k) = B(n, k, n) in the notation of Straub, equation 24.
%F The supercongruences T(n*p^r, k*p^r) == T(n*p^(r-1), k*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and k.
%F The formula T(n,k) = hypergeom([n+1, -n, -k], [1, 1], 1) allows the table indexing to be extended to negative values of n and k; clearly, we find that T(-n,k) = T(n-1,k) for all n and k. It appears that T(n,-k) = (-1)^n*T(n,k-1) for n >= 0, while T(n,-k) = (-1)^(n+1)*T(n,k-1) for n <= -1 [added Sep 10 2023: these follow from the identities immediately below]. (End)
%F T(n,k) = Sum_{i = 0..n} (-1)^(n+i) * binomial(n, i)*binomial(n+i, i)*binomial(k+i, i) = (-1)^n * hypergeom([n + 1, -n, k + 1], [1, 1], 1). - _Peter Bala_, Sep 10 2023
%F From _G. C. Greubel_, Oct 05 2023: (Start)
%F Let t(n,k) = T(n-k, k) (antidiagonals).
%F t(n, k) = Hypergeometric3F2([k-n, -k, n-k+1], [1,1], 1).
%F T(n, 2*n) = A363867(n).
%F T(3*n, n) = A363868(n).
%F T(2*n, 2*n) = A363869(n).
%F T(n, 3*n) = A363870(n).
%F T(2*n, 3*n) = A363871(n). (End)
%F T(n, k) = Sum_{i = 0..n} binomial(n, i)*binomial(n+i, i)*binomial(k, i). - _Peter Bala_, Feb 26 2024
%F Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*T(n, k) = A005259(n), the Apéry numbers associated with zeta(3). - _Peter Bala_, Jul 18 2024
%F From _Peter Bala_, Sep 21 2024: (Start)
%F Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*T(n, k) = binomial(2*n, n) = A000984(n).
%F Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*T(n-1, n-k) = A376458(n).
%F Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*T(i, k) = A143007(n, i). (End)
%F From _Peter Bala_, Oct 12 2024: (Start)
%F The square array = A063007 * transpose(A007318).
%F Conjecture: for positive integer m, Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k) * T(m*n, k) = ((m+1)*n)!/( ((m-1)*n)!*n!^2) (verified up to m = 10 using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). (End)
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ... A000012;
%e 1, 3, 5, 7, 9, 11, 13, ... A005408;
%e 1, 7, 19, 37, 61, 91, 127, ... A003215;
%e 1, 13, 55, 147, 309, 561, 923, ... A005902;
%e 1, 21, 131, 471, 1251, 2751, 5321, ... A008384;
%e 1, 31, 271, 1281, 4251, 11253, 25493, ... A008386;
%e 1, 43, 505, 3067, 12559, 39733, 104959, ... A008388;
%e 1, 57, 869, 6637, 33111, 124223, 380731, ... A008390;
%e 1, 73, 1405, 13237, 79459, 350683, 1240399, ... A008392;
%e 1, 91, 2161, 24691, 176251, 907753, 3685123, ... A008394;
%e 1, 111, 3191, 43561, 365751, 2181257, ... ... A008396;
%e ...
%e As a triangle:
%e [0] 1
%e [1] 1, 1
%e [2] 1, 3, 1
%e [3] 1, 7, 5, 1
%e [4] 1, 13, 19, 7, 1
%e [5] 1, 21, 55, 37, 9, 1
%e [6] 1, 31, 131, 147, 61, 11, 1
%e [7] 1, 43, 271, 471, 309, 91, 13, 1
%e [8] 1, 57, 505, 1281, 1251, 561, 127, 15, 1
%e [9] 1, 73, 869, 3067, 4251, 2751, 923, 169, 17, 1
%e ...
%e Inverse binomial transform of rows yield rows of triangle A063007:
%e 1;
%e 1, 2;
%e 1, 6, 6;
%e 1, 12, 30, 20;
%e 1, 20, 90, 140, 70;
%e 1, 30, 210, 560, 630, 252; ...
%e Product of the g.f. of row n and (1-x)^(n+1) generates the symmetric triangle A008459:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 9, 9, 1;
%e 1, 16, 36, 16, 1;
%e 1, 25, 100, 100, 25, 1;
%e ...
%p T := (n,k) -> binomial(n, k)*hypergeom([-k, k - n, k - n], [1, -n], 1):
%p seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # _Peter Luschny_, Feb 10 2018
%t T[n_, k_]:= HypergeometricPFQ[{-n, -k, n+1}, {1, 1}, 1] (* _Michael Somos_, Jun 03 2012 *)
%o (PARI) T(n,k)=sum(i=0,k,binomial(n,i)^2*binomial(n+k-i,k-i))
%o (Magma)
%o T:= func< n,k | (&+[Binomial(n,j)^2*Binomial(n+k-j,k-j): j in [0..k]]) >; // array
%o A108625:= func< n,k | T(n-k,k) >; // antidiagonals
%o [A108625(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 05 2023
%o (SageMath)
%o def T(n,k): return sum(binomial(n,j)^2*binomial(n+k-j, k-j) for j in range(k+1)) # array
%o def A108625(n,k): return T(n-k, k) # antidiagonals
%o flatten([[A108625(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 05 2023
%Y Rows include: A003215 (row 2), A005902 (row 3), A008384 (row 4), A008386 (row 5), A008388 (row 6), A008390 (row 7), A008392 (row 8), A008394 (row 9), A008396 (row 10).
%Y Cf. A063007, A099601 (n-th term of A_{2n} lattice), A108553.
%Y Cf. A008459 (h-vectors type B associahedra), A145904, A145905.
%Y Cf. A005258 (main diagonal), A108626 (antidiagonal sums).
%Y Cf. A108628, A208675, A363867, A363868, A363869, A363870, A363871.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jun 12 2005