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Search: id:A073405
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| A073405 |
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Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073406. |
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+0
6
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| 1, 36, 12, 1536, 888, 120, 80448, 62592, 15168, 1152, 5068800, 4813056, 1600704, 222336, 10944, 375598080, 413351424, 169917696, 32811264, 2992896, 103680, 32103751680, 39661608960, 19066503168, 4592982528
(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 row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...
The k-th convolution of U0(n) := A002605(n), n>= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*12^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073406(k,m).
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LINKS
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W. Lang First 7 rows .
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FORMULA
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Recursion for row polynomials defined in the comments: p(k, n)= 2*(2*(n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n)); q(k, n)= 4*((n+1)*p(k-1, n+1)+(n+2*(k+1))*q(k-1, n)), k >= 1.
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EXAMPLE
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k=2: U2(n)=2*((36+12*n)*(n+1)*U0(n+1)+(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
1; 36,12; 1536,888,120; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
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CROSSREFS
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Cf. A002605, A073387, A073406, A073403.
Sequence in context: A020340 A109256 A066583 this_sequence A056770 A061038 A058231
Adjacent sequences: A073402 A073403 A073404 this_sequence A073406 A073407 A073408
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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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