A000439 Powers of rooted tree enumerator. (Formerly M4608 N1965)

9, 30, 69, 133, 230, 369, 560, 814, 1143, 1560, 2079, 2715, 3484, 4403, 5490, 6764, 8245, 9954, 11913, 14145, 16674, 19525, 22724, 26298, 30275, 34684, 39555, 44919, 50808, 57255, 64294, 71960, 80289, 89318, 99085, 109629, 120990, 133209, 146328, 160390, 175439, 191520, 208679

Offset

1,1

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..43.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (n^4 + 18*n^3 + 83*n^2 + 114*n)/24. - Philippe Deléham, Feb 13 2004

G.f.: (2*x^3 - 9*x^2 + 15*x - 9)/(x - 1)^5. - Jinyuan Wang, Mar 17 2020

Maple

A000439:=(2*z-3)*(z**2-3*z+3)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[(n^4 + 18 n^3 + 83 n^2 + 114 n) / 24, {n, 50}] (* Vincenzo Librandi, Mar 18 2020 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {9, 30, 69, 133, 230}, 50] (* Harvey P. Dale, May 15 2023 *)

Prog

(PARI) a(n) = (n^4 + 18*n^3 + 83*n^2 + 114*n)/24; \\ Jinyuan Wang, Mar 17 2020
(Magma) [(n^4 + 18*n^3 + 83*n^2 + 114*n)/24: n in [1..50]]; // Vincenzo Librandi, Mar 18 2020
(Python)
def a(n): return (n**4 + 18*n**3 + 83*n**2 + 114*n)//24
print([a(n) for n in range(1, 44)]) # Michael S. Branicky, Sep 30 2021

Crossrefs

Sequence in context: A225275 A005919 A084370 * A002414 A273604 A273640

Adjacent sequences: A000436 A000437 A000438 * A000440 A000441 A000442

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Joerg Arndt, May 09 2013

Status

approved