A188111 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).
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
Offset: 1
Examples
Triangle starts:
1;
1, 1;
3, 2, 1;
10, 7, 3, 1;
38, 26, 12, 4, 1;
154, 105, 49, 18, 5, 1;
...
Links
- Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
Programs
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Maple
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
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 *) -
Maxima
T(n, m):=if n<=0 or m<=0 then 0 else if n
Vladimir Kruchinin, Apr 21 2016 */
Formula
T(n,m) = A037027(2*n-m-1,n-1)*m/n.
T(n,m) = if n<0 or m<0 or nVladimir 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