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Search: id:A111846
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| A111846 |
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Number of partitions of 4^n - 1 into powers of 4, also equals column 0 of triangle A111845, which shifts columns left and up under matrix 4-th power. |
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3
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| 1, 1, 4, 40, 1040, 78240, 18504256, 14463224448, 38544653734144, 357896006503348736, 11766320092785122862080, 1387031702368547767793690624, 592262859312707222259571097997312
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n) equals the partitions of 4^n-1 into powers of 4, or, the coefficient of x^(4^n-1) in 1/Product_{j>=0}(1-x^(4^j)).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..35
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FORMULA
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G.f.: A(x) = 1 + Sum_{n>=1} (1/n!)*Product_{j=0..n-1} L(4^j*x) where L(x) satisfies: x = Sum_{n>=1} -(-1)^n/n!*Product_{j=0..n-1} L(4^j*x); L(x) equals the g.f. of column 0 of the matrix log of P (A111849).
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EXAMPLE
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G.f. A(x) = 1 + L(x) + L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3!
+ L(x)*L(4*x)*L(4^2*x)*L(4^3*x)/4! + ...
where L(x) satisfies:
x = L(x) - L(x)*L(4*x)/2! + L(x)*L(4*x)*L(4^2*x)/3! -+ ...
and L(x) = x + 4/2!*x^2 + 56/3!*x^3 + 1728/4!*x^4 +....(A111849).
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PROGRAM
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(PARI) {a(n, q=4)=local(A=Mat(1), B); if(n<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, if(j==1, B[i, j]=(A^q)[i-1, 1], B[i, j]=(A^q)[i-1, j-1])); )); A=B); return(A[n+1, 1]))}
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CROSSREFS
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Cf. A111845 (triangle).
Cf. A002449
Sequence in context: A012957 A012977 A013108 this_sequence A102922 A139688 A072445
Adjacent sequences: A111843 A111844 A111845 this_sequence A111847 A111848 A111849
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2005
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