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

%I A001794 M4405 N1859 #117 Jun 09 2025 00:22:26
%S A001794 1,7,32,120,400,1232,3584,9984,26880,70400,180224,452608,1118208,
%T A001794 2723840,6553600,15597568,36765696,85917696,199229440,458752000,
%U A001794 1049624576,2387607552,5402263552,12163481600,27262976000,60850962432
%N A001794 Negated coefficients of Chebyshev T polynomials: [x^n](-T(n+6, x)), n >= 0.
%C A001794 A negated subdiagonal of A053120.
%C A001794 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
%C A001794 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
%C A001794 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
%D A001794 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.
%D A001794 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001794 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001794 T. D. Noe, <a href="/A001794/b001794.txt">Table of n, a(n) for n = 0..500</a>
%H A001794 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A001794 Michael Albert, Mike Atkinson and Robert Brignall, <a href="https://doi.org/10.37236/2442">The enumeration of three pattern classes using monotone grid classes</a>, E. J. Combinat., Vol. 19, No. 3 (2012), Article P20, chapter 5.4.1.
%H A001794 Takayuki Hibi, Nan Li and Hidefumi Ohsugi, <a href="http://arxiv.org/abs/1309.5155">The face vector of a half-open hypersimplex</a>, arXiv preprint arXiv:1309.5155 [math.CO], 2013-2014.
%H A001794 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A001794 C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, <a href="https://doi.org/10.1017/S0080454100006579">Tables of Chebyshev polynomials</a>, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
%H A001794 John Machacek, <a href="https://arxiv.org/abs/2410.06177">A finite totally nonnegative Grassmannian</a>, arXiv:2410.06177 [math.CO], 2024. See p. 11.
%H A001794 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001794 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A001794 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H A001794 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F A001794 a(n) = 2^(n-2)*(n+1)*(n+2)*(n+6)/3. [See a comment in A053120 on subdiagonals. - _Wolfdieter Lang_, Jan 03 2020]
%F A001794 G.f.: (1-x)/(1-2*x)^4. - _Simon Plouffe_ in his 1992 dissertation
%F A001794 a(n) = Sum_{k=0..floor((n+6)/2)} C(n+6, 2*k)*C(k, 3). - _Paul Barry_, May 15 2003
%F A001794 With a leading zero, the binomial transform of A000330. - _Paul Barry_, Jul 19 2003
%F A001794 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]
%F A001794 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
%F A001794 a(n) = (2^(n-1)/3)*binomial(n+2,2)*(n+6). - _Brad Clardy_, Mar 08 2012
%F A001794 E.g.f.: (1/3)*exp(2*x)*(3 + 15*x + 12*x^2 + 2*x^3). - _Stefano Spezia_, Jan 03 2020
%F A001794 From _Amiram Eldar_, Jan 05 2022: (Start)
%F A001794 Sum_{n>=0} 1/a(n) = 156*log(2)/5 - 511/25.
%F A001794 Sum_{n>=0} (-1)^n/a(n) = 241/25 - 108*log(3/2)/5. (End)
%F A001794 E.g.f.: exp(2*x)*(x^2/2 + x^3/6) (with two leading zeros). - _Enrique Navarrete_, Jun 03 2025
%p A001794 [seq(coeftayl((1-x)/(1-2*x)^4, x = 0, k), k=0..25)]; # _Muniru A Asiru_, Mar 20 2018
%t A001794 a[n_] := 2^(n-2)*(n+1)*(n+2)*(n+6)/3; a /@ Range[0, 20] (* _Giovanni Resta_, Mar 25 2017 *)
%t A001794 LinearRecurrence[{8,-24,32,-16},{1,7,32,120},30] (* _Harvey P. Dale_, Oct 08 2024 *)
%o A001794 (Magma) [2^(n-1)/3*Binomial(n+2,2)*(n+6) : n in [0..25]]; // _Brad Clardy_, Mar 08 2012
%o A001794 (PARI) a(n) = sum(i=0, n+1, sum(k=0, i, k^2*binomial(n+1, i))); \\ _Michel Marcus_, Mar 25 2017
%o A001794 (PARI) a(n) = - polcoeff(polchebyshev(n+6), n); \\ _Michel Marcus_, Mar 20 2018
%o A001794 (GAP) List([0..25],n->2^(n-2)*(n+1)*(n+2)*(n+6)/3); # _Muniru A Asiru_, Mar 20 2018
%Y A001794 Cf. A039991 (negative of column 6), A028297, A008310, A053120.
%Y A001794 With alternating signs, the o.g.f. (with offset 1) is the inverse of the o.g.f. of A065097.
%Y A001794 Cf. A001789 (partial sums), A081279 (binomial transform), A005900 (0 followed by inverse binomial transform).
%K A001794 nonn,easy
%O A001794 0,2
%A A001794 _N. J. A. Sloane_
%E A001794 Name clarified by _Wolfdieter Lang_, Nov 26 2019