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A108625 Square array, read by antidiagonals, where row n equals the crystal ball sequence for A_n lattice. 24
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, 1, 43, 271, 471, 309, 91, 13, 1, 1, 57, 505, 1281, 1251, 561, 127, 15, 1, 1, 73, 869, 3067, 4251, 2751, 923, 169, 17, 1, 1, 91, 1405, 6637, 12559, 11253, 5321, 1415, 217, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

From Peter Bala, Jul 18 2008 (Start):

Row reverse 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). - Peter Bala, Oct 28 2008

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux 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

FORMULA

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).

From Peter Bala, 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)

T(n, k) = hypergeom([-n, -k, n + 1], [1, 1], 1). - Michael Somos, Jun 03 2012

T(n, k) = binomial(n, k)*hypergeom([-k, k - n, k - n], [1, -n], 1). - Peter Luschny, Feb 10 2018

EXAMPLE

Square array begins:

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

  1,  3,    5,     7,     9,     11,      13, ...

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

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

  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, ...

As a triangle:

[0]  1

[1]  1,  1

[2]  1,  3,   1

[3]  1,  7,   5,    1

[4]  1, 13,  19,    7,    1

[5]  1, 21,  55,   37,    9,    1

[6]  1, 31, 131,  147,   61,   11,   1

[7]  1, 43, 271,  471,  309,   91,  13,   1

[8]  1, 57, 505, 1281, 1251,  561, 127,  15,  1

[9]  1, 73, 869, 3067, 4251, 2751, 923, 169, 17, 1

     ...

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;

  ...

MAPLE

T := (n, k) -> binomial(n, k)*hypergeom([-k, k - n, k - n], [1, -n], 1):

seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Feb 10 2018

MATHEMATICA

T[ n_, k_] := HypergeometricPFQ[ {-n, -k, n + 1}, {1, 1}, 1] (* Michael Somos, Jun 03 2012 *)

PROG

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

CROSSREFS

T(n, n-1) = A208675(n). T(n+1, n) = A108628(n). - Michael Somos, Jun 03 2012

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.

Sequence in context: A112996 A205099 A136621 * A177992 A112857 A118801

Adjacent sequences:  A108622 A108623 A108624 * A108626 A108627 A108628

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jun 12 2005

STATUS

approved

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Last modified October 18 15:21 EDT 2019. Contains 328162 sequences. (Running on oeis4.)