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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*diff(p(n),x)
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2
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Constant terms of polynomials related to Ramanujan psi polynomials (see Zeng reference).
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REFERENCES
| B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.
P. W. Shor, Problem 78-6: A combinatorial identity, in Problems and Solutions column, SIAM Review; problem in 20, p. 394 (1978); solution in 21, pp. 258-260 (1979). [From N. Sato (nsato7(AT)yahoo.ca), Feb 19 2010]
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LINKS
| S. Ramanujan, Notebook entry
P. W. Shor, A = B (but not quite); 3-d array with multiple recurrences
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.
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FORMULA
| 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].
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EXAMPLE
| Triangle begins
1;
1, 3;
2, 10, 15;
6, 40, 105, 105;
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
| 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}]] (* From Jean-François Alcover, May 31 2011 *)
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PROG
| (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
| Cf. A000142, A000312, A001147, A054589.
Sequence in context: A095675 A006743 A091811 * A025520 A099946 A011953
Adjacent sequences: A075853 A075854 A075855 * A075857 A075858 A075859
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KEYWORD
| nonn,tabl
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AUTHOR
| Frederic Chapoton (chapoton(AT)math.uqam.ca), Oct 15 2002
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