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Search: id:A104033
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| A104033 |
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Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2*n+1,2*k+1). |
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+0
3
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| 1, -3, 1, 25, -10, 1, -427, 175, -21, 1, 12465, -5124, 630, -36, 1, -555731, 228525, -28182, 1650, -55, 1, 35135945, -14449006, 1782495, -104676, 3575, -78, 1, -2990414715, 1229758075, -151714563, 8912475, -305305, 6825, -105, 1, 329655706465, -135565467080, 16724709820, -982532408
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OFFSET
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0,2
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COMMENT
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Column 0 equals signed A009843 (expansion of x/cosh(x)). Row sums form signed A000182 (expansion of tanh(x)).
The matrix logarithm is L(n,k)=-(-1)^(n-k)*A000182(n-k)*A103327(n,k), where A000182 = tangent numbers.
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FORMULA
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Column k: Sum_{j=0..n} C(2*n+1, 2*j+1)*T(j, k) = 0 (n>k), or 1 (n=k). Row n: Sum_{j=0..n} T(n, j)*C(2*j+1, 2*k+1) = 0 (k<n), or 1 (k=n). Sum_{k=0..n} T(n, k)*4^k = 1 for n>=0.
T(n, k) = (-1)^(n-k)*A000364(n-k)*A103327(n, k), where A000364 = Euler numbers.
Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n, k) = A002084(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2005
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EXAMPLE
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Rows begin:
1;
-3,1;
25,-10,1;
-427,175,-21,1;
12465,-5124,630,-36,1;
-555731,228525,-28182,1650,-55,1;
35135945,-14449006,1782495,-104676,3575,-78,1; ...
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PROGRAM
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(PARI) {T(n, k)=if(n<k|k<0, 0, ((matrix(n+1, n+1, m, j, if(m>=j, binomial(2*m-1, 2*j-1))))^-1)[n+1, k+1])}
(PARI) {T(n, k)=binomial(2*n+1, 2*k+1)* polcoeff(1/cosh(x+x*O(x^(2*n))), 2*n-2*k)*(2*n-2*k)!}
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CROSSREFS
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Cf. A000364, A103327, A009843, A000182.
Sequence in context: A137330 A138654 A072271 this_sequence A098815 A033464 A113099
Adjacent sequences: A104030 A104031 A104032 this_sequence A104034 A104035 A104036
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KEYWORD
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sign,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Feb 28 2005
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EXTENSIONS
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Edited by njas at the suggestion of Andrew Plewe, Jun 08 2007
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