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A075856
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Triangle formed from coefficients of the polynomials p(1)=x, p(n+1) = (n + x*(n+1))*p(n) + x*x*(d/dx)p(n).
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6
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1, 1, 3, 2, 10, 15, 6, 40, 105, 105, 24, 196, 700, 1260, 945, 120, 1148, 5068, 12600, 17325, 10395, 720, 7848, 40740, 126280, 242550, 270270, 135135, 5040, 61416, 363660, 1332100, 3213210, 5045040, 4729725, 2027025
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OFFSET
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1,3
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COMMENTS
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Constant terms of polynomials related to Ramanujan psi polynomials (see Zeng reference).
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LINKS
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FORMULA
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T(n, k) = (n-1) * T(n-1, k) + (n+k-1) * T(n-1, k-1). - Michael Somos, Mar 17 2011
G.f. A(x, t) = Sum_{n>0} p[n] t^n / n! satisfies (dA / dt) * (x + t - 1) = x * (1 + A)^2 * (x * (1 + A) - 1). - Michael Somos, Mar 17 2011
T(n, 1) = (n-1)! = A000142(n-1). T(n, n) = A001147(n). Sum_{k>0} T(n, k) = n^n = A000312(n). Sum_{k>0} T(n, k) x^k = p[n].
This triangle is A185164 read by diagonals.
Let F(x) = x + (1-x)*log(1-x). The e.g.f. is given by the compositional inverse
(x - t*F(x))^(-1) = x + t*x^2/2! + (t + 3*t^2)x^3/3! + (2*t + 10*t^2 + 15*t^3)*x^4/4! + ....
Let f(x) = 1/log(1+x) and define inductively D^(n+1)(f(x)) = f(x)*(d/dx)(D^n(f(x))) with D^(0)f(x) = f(x). Then D^(n)f = (-1)^n*sum {k = 1..n} T(n,k)*f^(n-k)/((1+x)^n*f^(2n+1)).
(End)
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EXAMPLE
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Triangle begins
1;
1, 3;
2, 10, 15;
6, 40, 105, 105;
24, 196, 700, 1260, 945;
120, 1148, 5068, 12600, 17325, 10395;
...
p(1) = x, p(2) = 3*x^2 + x, p(3) = 15*x^3 + 10*x^2 + 2*x, etc. - Michael Somos, Mar 17 2011
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MATHEMATICA
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p[1] = x; p[n_] := p[n] = (n - 1 + x*n)*p[n - 1] + x*x*D[p[n - 1], x]; Flatten[Rest[CoefficientList[#1, x]] & /@ Table[p[n], {n, 8}]] (* Jean-François Alcover, May 31 2011 *)
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PROG
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(PARI) {T(n, k) = if( k<1 || n<k, 0, if( n == 1, 1, (n-1) * T(n-1, k) + (n+k-1) * T(n-1, k-1)))}; /* Michael Somos, Mar 17 2011 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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