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A054335
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A convolution triangle of numbers based on A000984 (central binomial coefficients of even order).
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11
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1, 2, 1, 6, 4, 1, 20, 16, 6, 1, 70, 64, 30, 8, 1, 252, 256, 140, 48, 10, 1, 924, 1024, 630, 256, 70, 12, 1, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1, 48620, 65536, 51480, 28672, 12012, 3840, 924, 160, 18, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(sqrt(1-4*z)-x*z).
The column sequences are for m=0..20: A000984, A000302 (powers of 4), A002457, A002697, A002802, A038845, A020918, A038846, A020920, A040075, A020922, A045543, A020924, A054337, A020926, A054338, A020928, A054339, A020930, A054340, A020932.
Riordan array (1/sqrt(1-4x),x/sqrt(1-4x)). [From Paul Barry (pbarry(AT)wit.ie), May 06 2009]
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FORMULA
| a(n, 2*k+1) = binomial(n-k-1, k)*4^(n-2*k-1), a(n, 2*k) = binomial(2*(n-k), n-k)*binomial(n-k, k)/binomial(2*k, k), k >= 0, n >= m >= 0; a(n, m) := 0 if n<m.
Column recursion: a(n, m)=2*(2*n-m-1)*a(n-1, m)/(n-m), n>m >= 0, a(m, m) := 1.
G.f. for column m: cbie(x)*((x*cbie(x))^m, with cbie(x) := 1/sqrt(1-4*x).
G.f.: 1/(1-xy-2x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), May 06 2009]
Sum_{k, k>=0} T(n,2k)*(-1)^k*A000108(k) = A000108(n+1). - DELEHAM Philippe, Jan 30 2012
Sum_{k, 0<=k<=floor(n/2)} T(n-k,n-2k) = A098615(n). - DELEHAM Philippe, Feb 01 2012
T(n,k) = 4*T(n-1,k) + T(n-2,k-2) for k>=1 . - DELEHAM Philippe, Feb 02 2012
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EXAMPLE
| 1;
2,1;
6,4,1;
20,16,6,1;
70,64,30,8,1;
...
Fourth row polynomial (n=3): p(3,x)= 20+16*x+6*x^2+x^3
Contribution from Paul Barry (pbarry(AT)wit.ie), May 06 2009: (Start)
Production matrix begins
2, 1,
2, 2, 1,
0, 2, 2, 1,
-2, 0, 2, 2, 1,
0, -2, 0, 2, 2, 1,
4, 0, -2, 0, 2, 2, 1,
0, 4, 0, -2, 0, 2, 2, 1,
-10, 0, 4, 0, -2, 0, 2, 2, 1,
0, -10, 0, 4, 0, -2, 0, 2, 2, 1 (End)
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MATHEMATICA
| Flatten[ CoefficientList[#1, x] & /@ CoefficientList[ Series[1/(Sqrt[1 - 4*z] - x*z), {z, 0, 9}], z]] (* From Jean-François Alcover, Sep 08 2011, after g.f. *)
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CROSSREFS
| Cf. A000984, A035324, A054336 . Row sums: A026671(n).
Sequence in context: A073387 A125693 A094527 * A110681 A117852 A080245
Adjacent sequences: A054332 A054333 A054334 * A054336 A054337 A054338
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KEYWORD
| easy,nice,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 13 2000
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