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A086810
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Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.
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17
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1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 9, 21, 14, 0, 1, 14, 56, 84, 42, 0, 1, 20, 120, 300, 330, 132, 0, 1, 27, 225, 825, 1485, 1287, 429, 0, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 0, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 0, 1, 54, 936, 7644, 34498, 91728
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Mirror image of triangle A133336 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2008]
Contribution from Tom Copeland, Oct 09 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = t
P(3,t) = t + 2 t^2
P(4,t) = t + 5 t^2 + 5 t^3)
P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4
The o.g.f. A(x,t)= {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)], (see Drake et al.).
B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.
Let h(x,t)= 1/(dB/dx)= (1-x)^2/[1+(1+t)*x*(x-2)]=1/[1-t[2x+3x^2+4x^3+...]], an o.g.f. in x for row polynomials in t of A181289. Then P(n,t), is given by (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). These results are a special case of A133437 with u(x,t)= B(x,t), i.e., u_1=1 and (u_n)=-t for n>1. See A001003 for t = 1. (End)
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LINKS
| Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7
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FORMULA
| Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deleham's operator defined in A084938.
For k>0, T(n, k) = binomial(n+k-1, n)*binomial(n+2k, k)/(n+k+1); T(0, 0) = 1 and T(n, 0) = 0 if n>0.
Sum_{k>=0} T(n, k)*2^k = A107841(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 26 2005
Sum_{ k>=0} T(n-k, k) = A005043(n) . - Philippe DELEHAM, May 30 2005
T(n, k) = A108263(n+k, k) . - Philippe DELEHAM, May 30 2005
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2007
Sum_{k, 0<=k<=n}T(n,k)*5^k*(-2)^(n-k) = A152601(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2008]
Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*3^(n-k) = A154825(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 17 2009]
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EXAMPLE
| 1; 0, 1; 0, 1, 2; 0, 1, 5, 5; 0, 1, 9, 21, 14; ...
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CROSSREFS
| Diagonals : A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281
The diagonals (except for A000007) are also the diagonals of A033282.
Row sums : A001003 (Schroeder numbers)
Cf. A033282, A084938.
Sequence in context: A085650 A201910 A109450 * A085838 A094456 A010028
Adjacent sequences: A086807 A086808 A086809 * A086811 A086812 A086813
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 05 2003
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