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Search: id:A046521
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| A046521 |
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Array T(i,j)=binomial(-1/2-i,j)(-4)^j, i,j>=0 read by antidiagonals going down. |
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+0
3
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| 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; listen)
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OFFSET
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0,2
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COMMENT
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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 W. Lang reference, p. 306.
As a number triangle, this is the Riordan array (1/sqrt(1-4x),x/(1-4x)). - Paul Barry (pbarry(AT)wit.ie), May 30 2005
The A- and Z- sequences for this Riordan matrix are (see the W. 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). W. Lang, Jun 01 2007.
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REFERENCES
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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.
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LINKS
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W. Lang: First 10 rows.
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FORMULA
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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(Z(j)*a(n-1,j),j=0..n-1), n>=1; a(0,0)=1.
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EXAMPLE
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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.
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PROGRAM
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(PARI) T(i, j)=if(i<0|j<0, 0, (2*i+2*j)!*i!/(2*i)!/(i+j)!/j!)
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CROSSREFS
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Columns for m=0..10 are A000984, A002457, A002802, A020918-32 (only even numbers). Row sums: A046748.
Sequence in context: A089231 A052296 A019538 this_sequence A104684 A060538 A110183
Adjacent sequences: A046518 A046519 A046520 this_sequence A046522 A046523 A046524
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KEYWORD
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nonn,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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