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Search: id:A073387
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| A073387 |
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Convolution triangle of A002605(n) (generalized (2,2)-Fibonacci), n>=0. |
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+0
16
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| 1, 2, 1, 6, 4, 1, 16, 16, 6, 1, 44, 56, 30, 8, 1, 120, 188, 128, 48, 10, 1, 328, 608, 504, 240, 70, 12, 1, 896, 1920, 1872, 1080, 400, 96, 14, 1, 2448, 5952, 6672, 4512, 2020, 616, 126, 16, 1, 6688, 18192, 23040
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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The g.f. for the row polynomials P(n,x) := sum(a(n,m)*x^m,m=0..n) is 1/(1-(2+x+2*z)*z). See Shapiro et al. reference and comment under A053121 for such convolution triangles.
The column sequences (without leading zeros) give: A002605, A073388-94, A073397-8 for m=0..9. Row sums give A007482.
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LINKS
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W. Lang, First 10 rows.
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FORMULA
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a(n, m)=2*(p(m-1, n-m)*(n-m+1)*a(n-m+1)+q(m-1, n-m)*(n-m+2)*a(n-m))/(m!*12^m), n>=m>=1, with a(n)=a(n, m=0) := A002605(n), else 0; p(k, n) := sum(A(k, l)*n^(k-l), l=0..k) and q(k, n) := sum(B(k, l)*n^(k-l), l=0..k) with the number triangles A(k, m) := A073403(k, m) and B(k, m) := A073404(k, m).
a(n, m)=sum((2^(n-m))*binomial(n-k, m)*binomial(n-m-k, k)*(1/2)^k, k=0..floor((n-m)/2)) if n>m, else 0.
a(n, m)=((n-m+1)*a(n, m-1)+2*(n+m)*a(n-1, m-1))/(6*m), n>=m>=1, a(n, 0)=A002605(n+1), else 0.
G.f. for column m (without leading zeros): 1/(1-2*x*(1+x))^(m+1), m>=0.
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EXAMPLE
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{1},{2,1},{6,4,1},... (lower triangular matrix a(n,m), n>=m>=0, else 0).
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CROSSREFS
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Sequence in context: A137594 A112360 A118040 this_sequence A125693 A094527 A054335
Adjacent sequences: A073384 A073385 A073386 this_sequence A073388 A073389 A073390
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 2, 2002
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