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A046521 Array T(i,j) = binomial(-1/2-i,j)*(-4)^j, i,j >= 0 read by antidiagonals going down. 22
1, 2, 1, 6, 6, 1, 20, 30, 10, 1, 70, 140, 70, 14, 1, 252, 630, 420, 126, 18, 1, 924, 2772, 2310, 924, 198, 22, 1, 3432, 12012, 12012, 6006, 1716, 286, 26, 1, 12870, 51480, 60060, 36036, 12870, 2860, 390, 30, 1, 48620, 218790, 291720, 204204, 87516, 24310 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Or, a triangle related to A000984 (central binomial) and A000302 (powers of 4).

This is an example of a Riordan matrix. See the Shapiro et al. reference quoted under A053121 and Notes 1 and 2 of the Wolfdieter Lang reference, p. 306.

As a number triangle, this is the Riordan array (1/sqrt(1-4x),x/(1-4x)). - Paul Barry, May 30 2005

The A- and Z- sequences for this Riordan matrix are (see the Wolfdieter Lang link under A006232 for the D. G. Rogers, D. Merlini et al. and R. Sprugnoli references on Riordan A- and Z-sequences with a summary): A-sequence [1,4,0,0,0,...] and Z-sequence 4+2*A000108(n)*(-1)^(n+1)=[2, 2, -4, 10, -28, 84, -264, 858, -2860, 9724, -33592, 117572, -416024, 1485800, -5348880, 19389690, -70715340, 259289580, -955277400, 3534526380], n >= 0. The o.g.f. for the Z-sequence is 4-2*c(-x) with the Catalan number o.g.f. c(x). - Wolfdieter Lang, Jun 01 2007

As a triangle, T(2n,n) is A001448. Row sums are A046748. Diagonal sums are A176280. - Paul Barry, Apr 14 2010

From Wolfdieter Lang, Aug 10 2017: (Start)

The row polynomials R(n, x) of Riordan triangles R = (G(x), F(x)), with F(x)= x*Fhat(x), belong to the class of Boas-Buck polynomials (see the reference). Hence they satisfy the Boas-Buck identity (we use the notation of Rainville, Theorem 50, p. 141):

  (E_x - n*1)*R(n, x) = -Sum_{k=0..n-1} (alpha(k)*1 + beta(k)*E_x)*R(n-1.k, x), for n >= 0, where E_x = x*d/dx (Euler operator). The Boas-Buck sequences are given by alpha(k) := [x^k] ((d/dx)log(G(x))) and beta(k) := [x^k] (d/dx)log(Fhat(x)).

This entails a recurrence for the sequence of column m of the Riordan triangle T, n > m >= 0: R(n, m) = (1/(n-m))*Sum_{k=m..n-1} (alpha(n-1-k) + m*beta(n-1-k))*T(k, m), with input T(m,m).

For the present case the Boas-Buck identity for the row polynomials is (E_x - n*1)*R(n, x) = -Sum_{k=0..n-1} 2^(2*k+1)*(1 + 2*E_x)*R(n-1-k, x), for n >= 0. For the ensuing recurrence for the columns m of the triangle T see the formula and example section. (End)

REFERENCES

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).

Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.

LINKS

Table of n, a(n) for n=0..50.

P. Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.

W. Lang, First 10 rows.

W. Lang, On polynomials related to derivatives of the generating function of Catalan numbers, Fib. Quart. 40,4 (2002) 299-313; T(n,m) is called B(n,m) there.

FORMULA

T(n, m) = binomial(2*n, n)*binomial(n, m)/binomial(2*m, m), n >= m >= 0.

G.f. for column m: ((x/(1-4*x))^m)/sqrt(1-4*x).

Recurrence from the A-sequence given above: a(n,m) = a(n-1,m-1) + 4*a(n-1,m), for n >= m >= 1.

Recurrence from the Z-sequence given above: a(n,0) = Sum_{j=0..n-1} Z(j)*a(n-1,j), n >= 1; a(0,0)=1.

As a number triangle, T(n,k) = C(2*n,n)*C(n,k)/C(2*k,k) = C(n-1/2,n-k)*4^(n-k). - Paul Barry, Apr 14 2010

One of three infinite families of integral factorial ratio sequences of height 1 (see Bober, Theorem 1.2). The other two are A007318 and A068555. The triangular array equals exp(S), where the infinitesimal generator S has [2,6,10,14,18,...] on the main subdiagonal and zeros elsewhere. Recurrence equation for the square array: T(n+1,k) = (k+1)/(4*n+2)*T(n,k+1). - Peter Bala, Apr 11 2012

T(n,k) = 4^(n-k)*A006882(2*n - 1)/(A006882(2*n - 2*k)*A006882(2*k - 1)) = 4^(n-k)*(2*n - 1)!!/((2*n - 2*k)!*(2*k - 1)!!). - Peter Bala, Nov 07 2016

Boas-Buck recurrence for column m, m > n >= 0: T(n, m) = (2*(2*m+1)/(n-m))*Sum_{k=m..n-1} 4^(n-1-k)*T(k, m), with input T(n, n) = 1. See a comment above. - Wolfdieter Lang, Aug 10 2017

EXAMPLE

Array begins:

  1,  2,   6,  20,   70, ...

  1,  6,  30, 140,  630, ...

  1, 10,  70, 420, 2310, ...

  1, 14, 126, 924, 6006, ...

Recurrence from A-sequence: 140 = a(4,1) = 20 + 4*30.

Recurrence from Z-sequence: 252 = a(5,0) = 2*70 + 2*140 - 4*70 + 10*14 - 28*1.

From Paul Barry, Apr 14 2010: (Start)

As a number triangle, T(n, m) begins:

n\k       0      1       2       3      4      5     6    7   8  9 10 ...

0:        1

1:        2      1

2:        6      6       1

3:       20     30      10       1

4:       70    140      70      14      1

5:      252    630     420     126     18      1

6:      924   2772    2310     924    198     22     1

7:     3432  12012   12012    6006   1716    286    26    1

8:    12870  51480   60060   36036  12870   2860   390   30   1

9:    48620 218790  291720  204204  87516  24310  4420  510  34  1

10:  184756 923780 1385670 1108536 554268 184756 41990 6460 646 38  1

... [Reformatted and extended by Wolfdieter Lang, Aug 10 2017]

Production matrix begins

      2, 1,

      2, 4, 1,

     -4, 0, 4, 1,

     10, 0, 0, 4, 1,

    -28, 0, 0, 0, 4, 1,

     84, 0, 0, 0, 0, 4, 1,

   -264, 0, 0, 0, 0, 0, 4, 1,

    858, 0, 0, 0, 0, 0, 0, 4, 1,

  -2860, 0, 0, 0, 0, 0, 0, 0, 4, 1 (End)

Boas-Buck recurrence for column m = 2, and n = 4: T(4, 2) = (2*(2*2+1)/2) * Sum_{k=2..3} 4^(3-k)*T(k, 2) = 5*(4*1 + 1*10) = 70. - Wolfdieter Lang, Aug 10 2017

MATHEMATICA

t[i_, j_] := If[i < 0 || j < 0, 0, (2*i + 2*j)!*i!/(2*i)!/(i + j)!/j!]; Flatten[Reverse /@ Table[t[n, k - n] , {k, 0, 9}, {n, k, 0, -1}]][[1 ;; 51]] (* Jean-Fran├žois Alcover, Jun 01 2011, after PARI prog. *)

PROG

(PARI) T(i, j)=if(i<0 || j<0, 0, (2*i+2*j)!*i!/(2*i)!/(i+j)!/j!)

CROSSREFS

Columns for m=0..10 are A000984, A002457, A002802, A020918-A020932 (only even numbers). Row sums: A046748. Cf. A007318, A068555.

Cf. A001147, A006882.

Sequence in context: A019538 A269646 A269336 * A104684 A060538 A260848

Adjacent sequences:  A046518 A046519 A046520 * A046522 A046523 A046524

KEYWORD

nonn,tabl,easy

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified February 25 03:06 EST 2018. Contains 299630 sequences. (Running on oeis4.)