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A188111
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Triangle T(n,m) read by rows, [A(x)]^m = Sum_{n>=m} T(n,m)*x^n, where A(x) satisfies A(x) = x/(1-A(x)-A(x)^2).
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0
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1, 1, 1, 3, 2, 1, 10, 7, 3, 1, 38, 26, 12, 4, 1, 154, 105, 49, 18, 5, 1, 654, 444, 210, 80, 25, 6, 1, 2871, 1944, 927, 363, 120, 33, 7, 1, 12925, 8734, 4191, 1672, 575, 170, 42, 8, 1, 59345, 40040, 19305, 7810, 2761, 858, 231, 52, 9, 1
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OFFSET
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1,4
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LINKS
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Table of n, a(n) for n=1..55.
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
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FORMULA
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T(n,m) = A037027(2*n-m-1,n-1)*m/n.
T(n,m) = Sum_{i=1..n-m+1} A001002(i-1)*T(n-i,m-1), m>0. T(n,1) = A001002(n-1).
T(n,m) = if n<0 or m<0 or n<m then 0 else if n=m then 1 else if m=0 then 0 else T(n-1,m-1)+T(n,m+1)+T(n,m+2). - Vladimir Kruchinin, Apr 21 2016
Assuming the range n>=0 and 0<=k<=n we have T(n,k) = C(2*n-k,n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n],-4)*(k+1)/(n+1)) for n>=1 and T(0,0) = 1. - Peter Luschny, Apr 25 2016
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EXAMPLE
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Triangle starts:
1;
1, 1;
3, 2, 1;
10, 7, 3, 1;
38, 26, 12, 4, 1;
154, 105, 49, 18, 5, 1;
...
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MAPLE
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T := (n, k) -> `if`(n=0, 1, binomial(2*n-k, n)*hypergeom([(k-n)/2, (k-n+1)/2], [k-2*n], -4)*(k+1)/(n+1)): seq(seq(simplify(T(n, k), k=0..n)), n=0..10); # Peter Luschny, Apr 25 2016
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MATHEMATICA
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T[n_, m_] := T[n, m] = Which[n <= 0 || m <= 0, 0, n < m, 0, n == m, 1, True, T[n-1, m-1] + T[n, m+1] + T[n, m+2]];
Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 03 2019, after Vladimir Kruchinin *)
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PROG
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(Maxima)
T(n, m):=if n<=0 or m<=0 then 0 else if n<m then 0 else if n=m then 1 else T(n-1, m-1)+T(n, m+1)+T(n, m+2); /* Vladimir Kruchinin, Apr 21 2016 */
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CROSSREFS
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Cf. A001002, A037027.
Sequence in context: A248036 A307214 A185967 * A102472 A267629 A101894
Adjacent sequences: A188108 A188109 A188110 * A188112 A188113 A188114
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KEYWORD
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nonn,tabl
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AUTHOR
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Vladimir Kruchinin, Mar 21 2011
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STATUS
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approved
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