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Search: id:A097181
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| A097181 |
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Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 8^n, where R_n(y) forms the initial (n+1) terms of g.f. A097182(y)^(n+1). |
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6
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| 1, 1, 14, 1, 21, 210, 1, 28, 378, 3220, 1, 35, 595, 6475, 49910, 1, 42, 861, 11396, 108402, 778596, 1, 49, 1176, 18326, 207074, 1791930, 12198004, 1, 56, 1540, 27608, 361018, 3647672, 29389492, 191682920, 1, 63, 1953, 39585, 587727, 6783147
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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Row sums form A097185. Diagonal is A097183. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097184, where the g.f.s satisfy: A097182(x*A097184(x)) = A097184(x).
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FORMULA
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G.f.: A(x, y) = 2*y/((1-16*x*y) + (2*y-1)*(1-16*x*y)^(7/8)). G.f.: A(x, y) = A097183(x*y)/(1 - x*A097184(x*y)).
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EXAMPLE
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Row polynomials evaluated at y=1/2 equals powers of 8:
8^1 = 1 + 14/2;
8^2 = 1 + 21/2 + 210/2^2;
8^3 = 1 + 28/2 + 378/2^2 + 3220/2^3;
8^4 = 1 + 35/2 + 595/2^2 + 6475/2^3 + 49910/2^4;
where A097182(y)^(n+1) has the same initial terms as the n-th row:
A097182(y) = 1 + 7*x + 21*x^2 + 21*x^3 - 63*x^4 - 231*x^5 -+...
A097182(y)^2 = 1 + 14y +...
A097182(y)^3 = 1 + 21y + 210y^2 +...
A097182(y)^4 = 1 + 28y + 378y^2 + 3220y^3 +...
A097182(y)^5 = 1 + 35y + 595y^2 + 6475y^3 + 49910y^4 +...
Rows begin with n=0:
[1],
[1,14],
[1,21,210],
[1,28,378,3220],
[1,35,595,6475,49910],
[1,42,861,11396,108402,778596],
[1,49,1176,18326,207074,1791930,12198004],
[1,56,1540,27608,361018,3647672,29389492,191682920],
[1,63,1953,39585,587727,6783147,62974371,479497491,3019005990],...
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PROGRAM
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(PARI) {T(n, k)=if(n==0, 1, if(k==0, 1, if(k==n, 2^n*(4^n-sum(j=0, n-1, T(n, j)/2^j)), polcoeff((Ser(vector(n, i, T(n-1, i-1)), x)+x*O(x^k))^((n+1)/n), k, x))))}
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CROSSREFS
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Cf. A097182, A097183, A097184, A097185, A097186, A097179.
Sequence in context: A040207 A036163 A040208 this_sequence A040209 A071713 A147716
Adjacent sequences: A097178 A097179 A097180 this_sequence A097182 A097183 A097184
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 03 2004
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