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A038622
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Triangular array that counts rooted polyominoes.
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29
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1, 2, 1, 5, 3, 1, 13, 9, 4, 1, 35, 26, 14, 5, 1, 96, 75, 45, 20, 6, 1, 267, 216, 140, 71, 27, 7, 1, 750, 623, 427, 238, 105, 35, 8, 1, 2123, 1800, 1288, 770, 378, 148, 44, 9, 1, 6046, 5211, 3858, 2436, 1296, 570, 201, 54, 10, 1, 17303, 15115, 11505, 7590, 4302, 2067, 825, 265
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The PARI program gives any row k and any n-th term for this triangular array in square or right triangle array format. - Randall L. Rathbun, Jan 20 2002
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)=2*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+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
Triangle read by rows = partial sums of A064189 terms starting from the right. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2008]
Column k has e.g.f. exp(x)*(Bessel_I(k,2x)+Bessel_I(k+1,2x)) [Paul Barry, Mar 8 2011]
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REFERENCES
| D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.
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FORMULA
| a(n, k) = a(n-1, k-1) + a(n-1, k) + a(n-1, k+1) for k>0, a(n, k) = 2*a(n-1, k) + a(n-1, k+1) for k=0.
Riordan array ((sqrt(1-2x-3x^2)+3x-1)/(2x(1-3x)),(1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1-x)/(1+x+x^2),x/(1+x+x^2)). First column is A005773(n+1). Row sums are 3^n (A000244). If L=A038622, then L*L' is the Hankel matrix for A005773(n+1), where L' is the transpose of L. - Paul Barry (pbarry(AT)wit.ie), Sep 18 2006
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EXAMPLE
| 1; 2,1; 5,3,1; 13,9,4,1; ...
Triangle begins
1,
2, 1,
5, 3, 1,
13, 9, 4, 1,
35, 26, 14, 5, 1,
96, 75, 45, 20, 6, 1,
267, 216, 140, 71, 27, 7, 1,
750, 623, 427, 238, 105, 35, 8, 1,
2123, 1800, 1288, 770, 378, 148, 44, 9, 1
Production matrix is
2, 1,
1, 1, 1,
0, 1, 1, 1,
0, 0, 1, 1, 1,
0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 1, 1, 1
[Paul Barry, Mar 8 2011]
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MATHEMATICA
| nmax = 10; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, _] = 0; t[_?Negative, _?Negative] = 0; t[n_, 0] := 2 t[n-1, 0] + t[n-1, 1]; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]](* From Jean-François Alcover, Nov 09 2011 *)
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PROG
| (PARI) s=[0, 1]; {A038622(n, k)=if(n==0, 1, t=(2*(n+k)*(n+k-1)*s[2]+3*(n+k-1)*(n+k-2)*s[1])/((n+2*k-1)*n); s[1]=s[2]; s[2]=t; t)}
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CROSSREFS
| Cf. A005774.
A064189 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2008]
Sequence in context: A047858 A125171 A048472 * A193954 A162997 A112339
Adjacent sequences: A038619 A038620 A038621 * A038623 A038624 A038625
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KEYWORD
| tabl,easy,nice,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), TORSTEN.SILLKE(AT)LHSYSTEMS.COM
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EXTENSIONS
| More terms from David W. Wilson (davidwwilson(AT)comcast.net)
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