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A181783
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Array described in comments to A053482, here read by increasing diagonals. See comments below.
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3
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 16, 21, 7, 1, 1, 1, 65, 142, 63, 11, 1, 1, 1, 326, 1201, 709, 151, 16, 1, 1, 1, 1957, 12336, 9709, 2521, 311, 22, 1, 1, 1, 13700, 149989, 157971, 50045, 7186, 575, 29, 1, 1, 1, 10960
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table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,9
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COMMENTS
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We denote by a(n,p) the number in row number n>=0 and column number p>=0. The recurrence which defines the array is a(n,p)=n(p-1)a(n-1,p)+a(n,p-1). The initials values are given by a(n,0)=1=a(0,p) for all n>=0 and p>=0.
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LINKS
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Table of n, a(n) for n=0..57.
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FORMULA
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If we consider the e.g.f Psi(p) of column number p we have: Psi(p)(z)=Psi(p-1)(z)/((1-(p-1)z)) with Psi(1)(z)=exp(z). Then Psi(p)(z)=exp(z)/Prod_{k=0..p-1}(1-kz). We conclude that a(n,p)=n!sum_{m=0..n} sum_{k=1}^{p-1}(-1)^{p-1-k}k^{m+p-2}/((n-m)!(k-1)!(p-1-k)!). It seems after the recurrence (and its proof) in A053482 that:
a(n,p) = -sum_{k=1}^{p-1}s1(p,p-k)n(n-1)...(n-p+1)a(n-k,p)+1 where s1(m,n) are the classical Stirling numbers of first kind.
a(n,1) = 1 for every n.
a(1,p) = 1+p(p-1)/2 for every p.
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EXAMPLE
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Array read row after row:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 4, 7, 11, 16, ...
1, 1, 5, 21, 63, 151, 311, ...
1, 1, 16, 142, 709, 2521, ...
1, 1, 65, 1201, 9709, ...
a(4,3)=1201.
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MAPLE
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A181783 := proc(n, k)
option remember;
if n =0 or k = 0 then
1;
else
n*(k-1)*procname(n-1, k)+procname(n, k-1) ;
end if;
end proc:
seq(seq(A181783(d-k, k), k=0..d), d=0..12) ; # R. J. Mathar, Mar 02 2016
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CROSSREFS
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Cf. A000522, A053482, A185106.
Sequence in context: A008326 A181196 A227578 * A121395 A275377 A219585
Adjacent sequences: A181780 A181781 A181782 * A181784 A181785 A181786
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Richard Choulet, Dec 23 2012
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EXTENSIONS
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Edited by N. J. A. Sloane, Dec 24 2012
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STATUS
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approved
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