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A035324
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A convolution triangle of numbers, generalizing Pascal's triangle A007318.
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21
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1, 3, 1, 10, 6, 1, 35, 29, 9, 1, 126, 130, 57, 12, 1, 462, 562, 312, 94, 15, 1, 1716, 2380, 1578, 608, 140, 18, 1, 6435, 9949, 7599, 3525, 1045, 195, 21, 1, 24310, 41226, 35401, 19044, 6835, 1650, 259, 24, 1, 92378, 169766, 161052, 97954, 40963, 12021, 2450
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OFFSET
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1,2
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COMMENTS
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Triangle T(n,k), 1 <= k <= n, given by (0, 3/1, 1/3, 5/3, 3/5, 7/5, 5/7, 9/7, 7/9, 11/9, 9/11, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 28 2012
Riordan array (1, c(x)/sqrt(1-4x)) where c(x) = g.f. for Catalan numbers A000108, first column (k = 0) omitted. - Philippe Deléham, Jan 28 2012
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LINKS
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FORMULA
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a(n+1, m) = 2*(2*n+m)*a(n, m)/(n+1) + m*a(n, m-1)/(n+1), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1;
G.f. for column m: ((x*c(x)/sqrt(1-4*x))^m)/x, where c(x) = g.f. for Catalan numbers A000108.
a(n, m) =: s2(3; n, m).
T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012
T(n,m) = Sum_{k=m..n} k*binomial(k-1,k-m)*2^(k-m)*binomial(2*n-k-1,n-k))/n. - Vladimir Kruchinin, Aug 07 2013
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EXAMPLE
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Triangle begins:
1;
3, 1;
10, 6, 1;
35, 29, 9, 1;
126, 130, 57, 12, 1;
462, 562, 312, 94, 15, 1;
Triangle (0, 3, 1/3, 5/3, 3/5, ...) DELTA (1,0,0,0,0,0, ...) has an additional first column (1,0,0,...).
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MATHEMATICA
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a[n_, m_] /; n >= m >= 1 := a[n, m] = 2*(2*(n-1) + m)*(a[n-1, m]/n) + m*(a[n-1, m-1]/n); a[n_, m_] /; n < m = 0; a[n_, 0] = 0; a[1, 1] = 1; Flatten[ Table[ a[n, m], {n, 1, 10}, {m, 1, n}]] (* Jean-François Alcover, Feb 21 2012, from first formula *)
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PROG
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(Haskell)
a035324 n k = a035324_tabl !! (n-1) !! (k-1)
a035324_row n = a035324_tabl !! (n-1)
a035324_tabl = map snd $ iterate f (1, [1]) where
f (i, xs) = (i + 1, map (`div` (i + 1)) $
zipWith (+) ((map (* 2) $ zipWith (*) [2 * i + 1 ..] xs) ++ [0])
([0] ++ zipWith (*) [2 ..] xs))
(Sage)
@cached_function
def T(n, k):
if n == 0: return n^k
return sum(binomial(2*i-1, i)*T(n-1, k-i) for i in (1..k-n+1))
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CROSSREFS
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Alternating row sums give A000108 (Catalan numbers).
If offset 0 (n >= m >= 0): convolution triangle based on A001700 (central binomial coeffs. of odd order).
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KEYWORD
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AUTHOR
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STATUS
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approved
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