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A099174
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Triangle read by rows: coefficients of modified Hermite polynomials.
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4
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1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0<=k<=n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 14 2006
Riordan array [exp(x^2/2),x]. [From Paul Barry (pbarry(AT)wit.ie), Nov 06 2008]
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LINKS
| A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory
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FORMULA
| h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials (A060821, A059343).
T(n,k)=n!/[2^((n-k)/2)*((n-k)/2)!k! ] if n-k>=0 is even; 0 otherwise. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 14 2006
G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Apr 10 2009]
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EXAMPLE
| h(0,x) = 1
h(1,x) = x
h(2,x) = x^2 + 1
h(3,x) = x^3 + 3*x
h(4,x) = x^4 + 6*x^2 + 3
h(5,x) = x^5 + 10*x^3 + 15*x
h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 06 2008: (Start)
Triangle begins
1,
0, 1,
1, 0, 1,
0, 3, 0, 1,
3, 0, 6, 0, 1,
0, 15, 0, 10, 0, 1,
15, 0, 45, 0, 15, 0, 1
Production array starts
0, 1,
1, 0, 1,
0, 2, 0, 1,
0, 0, 3, 0, 1,
0, 0, 0, 4, 0, 1,
0, 0, 0, 0, 5, 0, 1 (End)
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MAPLE
| T:=proc(n, k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 14 2006
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CROSSREFS
| Row sums (values at 1) are A000085. Values at 2 are A005425.
Sequence in context: A126595 A179898 A066325 * A137297 A178117 A095710
Adjacent sequences: A099171 A099172 A099173 * A099175 A099176 A099177
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KEYWORD
| nonn,tabl
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AUTHOR
| Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004
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