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A090452
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Scaled array A078740 ((3,2)-Stirling2).
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9
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1, 1, 3, 2, 1, 7, 16, 15, 5, 1, 12, 51, 105, 114, 63, 14, 1, 18, 118, 396, 771, 910, 644, 252, 42, 1, 25, 230, 1110, 3235, 6083, 7580, 6240, 3270, 990, 132, 1, 33, 402, 2600, 10365, 27483, 50464, 65331, 59625, 37620, 15642, 3861, 429, 1, 42, 651, 5390, 27825, 97188
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| This scaled Stirling2 array will be called s2_{3,2}(n,m).
The sequence of row lengths is [1,3,5,7,...]=A005408(n-1).
The generating function for the sequence from column nr. m is G(m,x)=(x^ceiling(m/2))*P(m,x)/(1-x)^(2*m-3) with the row polynomials of array A091029(m,k).
The generating functions of the column sequences obey the hypergeometric differential-difference eq.:x*(1-x)*G''(m,x) + 2*(1-m*x)*G'(m,x) - m*(m-1)*G(m,x) = 2*m*x*G'(m-1,x) + 2*m*(m-1)*G(m-1,x) + m*(m-1)*G(m-2,x), m>=3; with G(2,x)=x/(1-x) and G(1,x)=0. The primes denote differentiation w.r.t. x.
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LINKS
| W. Lang, First 8 rows.
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FORMULA
| a(n, m)=(m!/((n+1)!*n!))*A078740(n, m), n>=1, 2<= m <=2*n.
Recursion: a(n, m)=((n+m-1)*(n+m-2)*a(n-1, m)+2*(n+m-2)*m*a(n-1, m-1)+m*(m-1)*a(n-1, m-2))/((n+1)*n), n>=2, 2<=m<=2*n, a(1, 2)=1, a(n, 0) := 0, a(n, 1) := 0 (from the recursion of array A078740).
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EXAMPLE
| [1]; [1,3,2]; [1,7,16,15,5]; [1,12,51,105,114,63,14]; ...
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CROSSREFS
| a(n, 2*n)=A000108(n) (Catalan), n>=1, a(n, 2*n-1)=3*A002054(n-1), n>=2, a(n, 2*n-2)=A091031(n), n>=2.
The column sequences (without leading zeros) are: A000012 (powers of 1), A055998, A090453-4, A091026-7, etc.
Cf. A090442 (row sums). The alternating row sums are 0 except for row n=1 which gives 1.
Sequence in context: A143774 A196842 A158474 * A193924 A110439 A065602
Adjacent sequences: A090449 A090450 A090451 * A090453 A090454 A090455
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 23 2003
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