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Search: id:A008298
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| A008298 |
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Triangle of D'Arcais numbers. |
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+0
7
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| 1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 144, 450, 215, 30, 1, 1440, 3394, 2475, 565, 45, 1, 5760, 30912, 28294, 9345, 1225, 63, 1, 75600, 293292, 340116, 147889, 27720, 2338, 84, 1, 524160, 3032208, 4335596, 2341332, 579369, 69552, 4074, 108, 1, 6531840, 36290736, 57773700, 38049920, 11744775, 1857513, 154350, 6630, 135, 1
(list; table; graph; listen)
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OFFSET
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1,2
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.
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FORMULA
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G.f.: Sum_{1<=k<=n} T(n, k)*u^k*t^n/n! = ((1-t)*(1-t^2)*(1-t^3)...)^(-u).
Recurrence for degree n D'Arcais polynomials T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = Sum_{k=1..n} (n-1)!/(n-k)!*sigma(k)*u*T(n-k; u), T(0; u) = 1. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 11 2002
T(n; u) = n!*Sum_{pi} Product_{i=1..n} binomial(u+k(i)-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Oct 11 2002
E.g.f.: exp(Sum_{n>0} sigma(n)*u*x^n/n), where sigma(n)=A000203(n). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 10 2003
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EXAMPLE
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exp(Sum_{n>0} sigma(n)*u*x^n/n) = 1+u*x/1!+(3*u+u^2)*x^2/2!+(8*u+9*u^2+u^3)*x^3/3!+(42*u+59*u^2+18*u^3+u^4)*x^4/4!+...
1; 3,1; 8,9,1; 42,59,18,1; ...
T(4; u) = 4!*(binomial(u+3,4)+binomial(u+1,2)*binomial(u,1)+binomial(u+1,2)+binomial(u,1)^2+binomial(u,1)) = 42*u+59*u^2+18*u^3+u^4.
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CROSSREFS
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Diagonals give A038048, A059356, A059357.
Row sums give A053529.
Sequence in context: A077897 A007023 A076238 this_sequence A039692 A071815 A120236
Adjacent sequences: A008295 A008296 A008297 this_sequence A008299 A008300 A008301
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KEYWORD
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nonn,tabl,nice,easy
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AUTHOR
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njas
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 28 2001
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