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A091965
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Triangle read by rows: T(n,k)=number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps).
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31
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1, 3, 1, 10, 6, 1, 36, 29, 9, 1, 137, 132, 57, 12, 1, 543, 590, 315, 94, 15, 1, 2219, 2628, 1629, 612, 140, 18, 1, 9285, 11732, 8127, 3605, 1050, 195, 21, 1, 39587, 52608, 39718, 19992, 6950, 1656, 259, 24, 1, 171369, 237129, 191754, 106644, 42498, 12177, 2457
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| T(n,0)=A002212(n+1), T(n,1)=A045445(n+1), Row sums give A026378.
The inverse is A123965. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 17 2006
Reversal of A084536 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 23 2007
Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=3*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+3*T(n-1,k)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007
5^n = (n-th row terms) dot (first n+1 terms in (1,2,3,...)). Example for row 4: 5^4 = 625 = (137, 132, 57, 12, 1) dot (1, 2, 3, 4, 5) = (137 + 264 + 171 + 48 + 5) = 625. - Gary W. Adamson, Jun 15 2011
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REFERENCES
| A. Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.
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FORMULA
| G.f.: G=2/(1-3*z-2*t*z+sqrt(1-6*z+5*z^2)). Alternatively, G=M/(1-t*z*M), where M=1+3*z*M+z^2*M^2.
Sum_{k, k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A002212(m+n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005
The triangle may also be generated from M^n * [1,0,0,0...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and [3,3,3...] in the main diagonal. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 17 2006
Sum_{k, 0<=k<=n}T(n,k)*(k+1)=5^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
Sum_{k, 0<=k<=n}T(n,k)*x^k = A117641(n), A033321(n), A007317(n), A002212(n+1), A026378(n+1) for x = -3, -2, -1, 0, 1 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2009]
T(n,k) = (k+1)*sum(m=k..n, (binomial(2*(m+1),m-k)*binomial(n,m))/(m+1)). [From Vladimir Kruchinin, Oct 08 2011]
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EXAMPLE
| Triangle begins:
[1],
[3, 1],
[10, 6, 1],
[36, 29, 9, 1],
[137, 132, 57, 12, 1],
[543, 590, 315, 94, 15, 1],
[2219, 2628, 1629, 612, 140, 18, 1]
T(3,1)=29 because we have UDU, UUD, 9 HHU paths, 9 HUH paths and 9 UHH paths.
Production matrix begins
3, 1
1, 3, 1
0, 1, 3, 1
0, 0, 1, 3, 1
0, 0, 0, 1, 3, 1
0, 0, 0, 0, 1, 3, 1
0, 0, 0, 0, 0, 1, 3, 1
0, 0, 0, 0, 0, 0, 1, 3, 1
0, 0, 0, 0, 0, 0, 0, 1, 3, 1
0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1
- From Philippe DELEHAM, Nov 07 2011
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MATHEMATICA
| nmax = 9; t[n_, k_] := ((k+1)*n!*Hypergeometric2F1[k+3/2, k-n, 2k+3, -4]) / ((k+1)!*(n-k)!); Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* From Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)
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PROG
| (Maxima)
T(n, k):=(k+1)*sum((binomial(2*(m+1), m-k)*binomial(n, m))/(m+1), m, k, n); [From Vladimir Kruchinin, Oct 08 2011]
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CROSSREFS
| Cf. A002212, A045445, A026378.
Cf. A123965.
Sequence in context: A134283 A035324 A171814 * A171568 A107056 A116384
Adjacent sequences: A091962 A091963 A091964 * A091966 A091967 A091968
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 13 2004
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