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Search: id:A107729
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| A107729 |
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Triangle T(n,k), 0<=k<=n, read by rows, defined by T(0,0) = 1; T(0,k) = 0 if k<0 or if k>0; T(n,k) = k*T(n-1,k-1) + (k+2)*T(n-1,k+1). |
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+0
1
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| 1, 0, 1, 2, 0, 2, 0, 8, 0, 6, 16, 0, 40, 0, 24, 0, 136, 0, 240, 0, 120, 272, 0, 1232, 0, 1680, 0, 720, 0, 3968, 0, 12096, 0, 13440, 0, 5040, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320, 0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880, 353792, 0
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Triangle is related to the tangent numbers A000182.
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REFERENCES
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S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 446.
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FORMULA
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T(n, n) = n!; T(n, 0) = 0 if n = 2m+1; T(n, 0) = A000182(m+1) if n = 2m.
Sum_{k, k>=0} T(m, k)*T(n, k)*(k+1) = T(m+n, 0).
Sum_{k, k>=0} T(n, k) = |A003707(n+1)|.
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EXAMPLE
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Triangle begins:
1;
0, 1;
2, 0, 2;
0, 8, 0, 6;
16, 0, 40, 0, 24;
0, 136, 0, 240, 0, 120;
272, 0, 1232, 0, 1680, 0, 720;
0, 3968, 0, 12096, 0, 13440, 0, 5040;
7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320;
0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880;
353792, 0, 3610112, 0, 12207360, 0, 18627840, 0, 13305660, 0, 3628800;
...
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MAPLE
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T:=proc(n, k) if k=-1 then 0 elif n=1 and k=1 then 1 elif k>n then 0 else (k-1)*T(n-1, k-1)+(k+1)*T(n-1, k+1) fi end: for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form (Deutsch) [Produces triangle with a different offset]
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CROSSREFS
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Similar to A104035. Leading edge is essentially A000182.
Cf. A003707.
Adjacent sequences: A107726 A107727 A107728 this_sequence A107730 A107731 A107732
Sequence in context: A047765 A068463 A099554 this_sequence A113400 A136668 A057498
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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njas, Jun 10 2005
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 13 2005
Additional comments from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2005
Edited by njas, Aug 23 2008 at the suggestion of R. J. Mathar
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