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Search: id:A136590
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| A136590 |
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Triangle of trinomial logarithmic coefficients: A027907(n,k) = Sum_{i=0..k} T(k,i)*n^i/k!. |
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+0
7
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| 1, 0, 1, 0, 1, 1, 0, -4, 3, 1, 0, 6, -13, 6, 1, 0, 24, -10, -25, 10, 1, 0, -240, 394, -135, -35, 15, 1, 0, 720, -2016, 1834, -525, -35, 21, 1, 0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1, 0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1, 0, 362880, -1155024
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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A027907 is the triangle of trinomial coefficients.
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FORMULA
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E.g.f. of column k = log(1 + x + x^2)^k / k! for k>=0. Central trinomial coefficients: A002426(n) = Sum_{k=0..n} T(n,k)*n^k/n!.
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 1, 1;
0, -4, 3, 1;
0, 6, -13, 6, 1;
0, 24, -10, -25, 10, 1;
0, -240, 394, -135, -35, 15, 1;
0, 720, -2016, 1834, -525, -35, 21, 1;
0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1;
0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1;
0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1; ...
Trinomial coefficients can be calculated as illustrated by:
A027907(4,3) = (T(3,0)*4^0 + T(3,1)*4^1 + T(3,2)*4^2 + T(3,3)*4^3)/3! =
(0 - 4*4 + 3*4^2 + 1*4^3)/3! = 96/6 = 16.
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PROGRAM
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(PARI) T(n, k)=n!/k!*polcoeff(log(1+x+x^2 +x*O(x^n))^k, n)
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CROSSREFS
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Cf. columns: A136591, A136592, A136593; A136594 (unsigned row sums); A136595 (matrix inverse); A027907, A002426.
Adjacent sequences: A136587 A136588 A136589 this_sequence A136591 A136592 A136593
Sequence in context: A131027 A133475 A021236 this_sequence A117026 A083904 A129810
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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), Jan 10 2008
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