|
| |
|
|
A001794
|
|
Negated coefficients of Chebyshev T polynomials: [x^n](-T(n+6, x)), n >= 0.
(Formerly M4405 N1859)
|
|
17
|
|
|
|
1, 7, 32, 120, 400, 1232, 3584, 9984, 26880, 70400, 180224, 452608, 1118208, 2723840, 6553600, 15597568, 36765696, 85917696, 199229440, 458752000, 1049624576, 2387607552, 5402263552, 12163481600, 27262976000, 60850962432
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
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
|
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].
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.
|
|
|
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]
a(n) = Sum_{k=0..floor((n+6)/2)} C(n+6, 2*k)*C(k, 3). - Paul Barry, May 15 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
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 *)
|
|
|
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
|
With alternating signs, the o.g.f. (with offset 1) is the inverse of the o.g.f. of A065097.
Cf. A001789 (partial sums), A081279 (binomial transform), A005900 (0 followed by inverse binomial transform).
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
