OFFSET
0,2
COMMENTS
A negated subdiagonal of A053120.
If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then a(n-2) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
The third corrector line for transforming 2^n offset 0 with a leading 1 into the Fibonacci sequence. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
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
T. D. Noe, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael Albert, Mike Atkinson and Robert Brignall, The enumeration of three pattern classes using monotone grid classes, E. J. Combinat., Vol. 19, No. 3 (2012), Article P20, chapter 5.4.1.
Takayuki Hibi, Nan Li and Hidefumi Ohsugi, The face vector of a half-open hypersimplex, arXiv preprint arXiv:1309.5155 [math.CO], 2013-2014.
Milan Janjic, Two Enumerative Functions.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
John Machacek, A finite totally nonnegative Grassmannian, arXiv:2410.06177 [math.CO], 2024. See p. 11.
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 linear recurrences with constant coefficients, signature (8,-24,32,-16).
FORMULA
a(n) = 2^(n-2)*(n+1)*(n+2)*(n+6)/3. [See a comment in A053120 on subdiagonals. - Wolfdieter Lang, Jan 03 2020]
G.f.: (1-x)/(1-2*x)^4. - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=0..floor((n+6)/2)} C(n+6, 2*k)*C(k, 3). - Paul Barry, May 15 2003
With a leading zero, the binomial transform of A000330. - Paul Barry, Jul 19 2003
a(n) = Sum_{i=0..n+1} (Sum{k=0..i} (k^2*binomial(n+1, i))). - Jon Perry, Feb 26 2004 [corrected by Michel Marcus, Mar 25 2017]
Binomial transform of a(n) = (2*n^3 + 6*n^2 + 7*n + 3)/3 offset 0. a(3)=120. - Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009
a(n) = (2^(n-1)/3)*binomial(n+2,2)*(n+6). - Brad Clardy, Mar 08 2012
E.g.f.: (1/3)*exp(2*x)*(3 + 15*x + 12*x^2 + 2*x^3). - Stefano Spezia, Jan 03 2020
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=0} 1/a(n) = 156*log(2)/5 - 511/25.
Sum_{n>=0} (-1)^n/a(n) = 241/25 - 108*log(3/2)/5. (End)
MAPLE
[seq(coeftayl((1-x)/(1-2*x)^4, x = 0, k), k=0..25)]; # Muniru A Asiru, Mar 20 2018
MATHEMATICA
a[n_] := 2^(n-2)*(n+1)*(n+2)*(n+6)/3; a /@ Range[0, 20] (* Giovanni Resta, Mar 25 2017 *)
LinearRecurrence[{8, -24, 32, -16}, {1, 7, 32, 120}, 30] (* Harvey P. Dale, Oct 08 2024 *)
PROG
(Magma) [2^(n-1)/3*Binomial(n+2, 2)*(n+6) : n in [0..25]]; // Brad Clardy, Mar 08 2012
(PARI) a(n) = sum(i=0, n+1, sum(k=0, i, k^2*binomial(n+1, i))); \\ Michel Marcus, Mar 25 2017
(PARI) a(n) = - polcoeff(polchebyshev(n+6), n); \\ Michel Marcus, Mar 20 2018
(GAP) List([0..25], n->2^(n-2)*(n+1)*(n+2)*(n+6)/3); # Muniru A Asiru, Mar 20 2018
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Name clarified by Wolfdieter Lang, Nov 26 2019
STATUS
approved