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Search: id:A073406
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| A073406 |
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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 A073405. |
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4
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| 2, 36, 12, 1056, 672, 96, 43968, 40416, 10752, 864, 2396160, 2815488, 1051776, 156672, 8064, 161879040, 226492416, 105981696, 22125312, 2121984, 76032, 13044326400, 20766633984, 11446769664, 2995605504
(list; table; graph; listen)
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OFFSET
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0,1
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COMMENT
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The row polynomials are q(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 p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073405(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*((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.
2; 36,12; 1056,672,96; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).
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CROSSREFS
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Cf. A002605, A073387, A073403-A073405.
Sequence in context: A094716 A094725 A095397 this_sequence A096513 A037418 A058517
Adjacent sequences: A073403 A073404 A073405 this_sequence A073407 A073408 A073409
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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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