A001794 Negated coefficients of Chebyshev T polynomials: [x^n](-T(n+6, x)), n >= 0.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

a(n-2) is the number of strings of length n defined on {0, 1, 2, 3} that have exactly two 2's and no 3's or exactly three 3's and no 2's. For example, for n=4, a(2)=32 since the strings are the 6 permutations of 2200, the 12 permutations of 2201, the 6 permutations of 2211, the 4 permutations of 3330, and the 4 permutations of 3331. - Enrique Navarrete, Jun 03 2025

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 sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Crossrefs

Cf. A039991 (negative of column 6), A028297, A008310, A053120.

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).

Programs

GAP

List([0..25],n->2^(n-2)*(n+1)*(n+2)*(n+6)/3); # Muniru A Asiru, Mar 20 2018

Magma

[2^(n-1)/3*Binomial(n+2,2)*(n+6) : n in [0..25]]; // Brad Clardy, Mar 08 2012

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 *)

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

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)

E.g.f.: exp(2*x)*(x^2/2 + x^3/6) (with two leading zeros). - Enrique Navarrete, Jun 03 2025

Extensions

Name clarified by Wolfdieter Lang, Nov 26 2019