0,5

Compare to the corresponding array A108553 of crystal ball sequences for D_n lattice.

Comment from Peter Bala (pbala(AT)toucansurf.com), Jul 18 2008 (Start): Transpose of A099608.

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).

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..inf} 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.

For the entries in column k we have zeta(2) = (1 + 1/4 + 1/9 + ... + 1/k^2) + 2*sum {n = 1..inf} (-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.

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..inf} (-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) - ...).

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..inf} (-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..inf} (-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) - ... .

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)

This array is the Hilbert transform of triangle A008459 (see A145905 for the definition of the Hilbert transform). [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

A. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3). An informal report. Math. Intelligencer 1 (1978/79), no 4, 195-203.

Eric Weisstein's World of Mathematics, Apery number

T(n, k) = Sum_{i=0..k} C(n, i)^2 * C(n+k-i, k-i). G.f. for row n: (Sum_{i=0..n} C(n, i)^2 * x^i)/(1-x)^(n+1).

Comments from Peter Bala (pbala@toucansurf.com), Jul 23 2008 (Start): O.g.f. row n: 1/(1-x)*Legendre_P(n,(1+x)/(1-x)). 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. Main diagonal is A005258.

Recurrence relations: 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,... ; 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,... ; 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,... .

Series acceleration formulas for zeta(2): Row n: zeta(2) = 2*(1 - 1/2^2 + 1/3^2 - ... + (-1)^(n+1)/n^2) + (-1)^n*sum {k = 1..inf} 1/(k^2*T(n,k-1)*T(n,k)); Column k: zeta(2) = 1 + 1/2^2 + 1/3^2 + ... + 1/k^2 + 2*sum {n = 1..inf} (-1)^(n+1)/(n^2*T(n-1,k)*T(n,k)); Main diagonal: zeta(2) = 5 * sum {n = 1..inf} (-1)^(n+1)/(n^2*T(n-1,n-1)*T(n,n)).

Conjectural result for superdiagonals: zeta(2) = 1+1/2^2 + ... +1/k^2 + sum {n = 1..inf} (-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... . Conjectural result for subdiagonals: zeta(2) = 2*(1-1/2^2 + ... +(-1)^(k+1)/k^2) + (-1)^k*sum {n = 1..inf} (-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... .

Conjectural congruences: The main superdiagonal numbers S(n):= T(n,n+1) appear to satisfy the super congruences S(m*p^r - 1) = S(m*p^(r-1) - 1) (mod p^(3*r)) for prime p greater than 3 and m,r in N. 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)

Square array begins:

1,1,1,1,1,1,1,1,1,1,1,...

1,3,5,7,9,11,13,15,17,19,21,...

1,7,19,37,61,91,127,169,217,...

1,13,55,147,309,561,923,1415,...

1,21,131,471,1251,2751,5321,...

1,31,271,1281,4251,11253,25493,...

1,43,505,3067,12559,39733,104959,...

1,57,869,6637,33111,124223,380731,...

1,73,1405,13237,79459,350683,1240399,...

Inverse binomial transform of rows yield

rows of triangle A063007:

1;

1,2;

1,6,6;

1,12,30,20;

1,20,90,140,70;

1,30,210,560,630,252; ...

Product of the g.f. of row n and (1-x)^(n+1)

generates the symmetric triangle A008459:

1;

1,1;

1,4,1;

1,9,9,1;

1,16,36,16,1;

1,25,100,100,25,1; ...

(PARI) T(n, k)=sum(i=0, k, binomial(n, i)^2*binomial(n+k-i, k-i))

Cf. A108553, A008459, A063007, A108626 (antidiagonal sums), A005258 (main diagonal), A099601 (n-th term of A_{2n} lattice), 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).

A008459 (h-vectors type B associahedra), A145904, A145905. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

Adjacent sequences: A108622 A108623 A108624 this_sequence A108626 A108627 A108628

Sequence in context: A128119 A112996 A136621 this_sequence A112857 A118801 A080936

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jun 12 2005