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A006974 Coefficients of Chebyshev T polynomials: a(n) = A053120(n+8, n), n >= 0.
(Formerly M4631)
14

%I M4631

%S 1,9,50,220,840,2912,9408,28800,84480,239360,658944,1770496,4659200,

%T 12042240,30638080,76873728,190513152,466944000,1133117440,2724986880,

%U 6499598336,15386804224,36175872000,84515225600,196293427200,453437816832

%N Coefficients of Chebyshev T polynomials: a(n) = A053120(n+8, n), n >= 0.

%C If X_1,X_2,...,X_n are 2-blocks of a (2n+1)-set X then, for n>=3, a(n-3) is the number of (n+4)-subsets of X intersecting each X_i, (i=1,2,...,n). - _Milan Janjic_, Nov 18 2007

%C The fourth 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]

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

%H 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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H M. H. Albert, M. D. Atkinson, R. Brignall, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p20">The enumeration of three pattern classes using monotone grid classes</a>, El. J. Combinat. 19 (3) (2012) P20, Chapter 5.4.1.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F G.f.: (1-x)/(1-2*x)^5.

%F a(n) = Sum_{k=0..floor((n+8)/2)} C(n+8, 2k)*C(k, 4). - _Paul Barry_, May 15 2003

%F Binomial transform of a(n)=(24*n^4-134*n^3+261*n^2-130*n+3)/3 offset 0. a(3)=220. [Al Hakanson (hawkuu(AT)gmail.com), Jun 01 2009]

%F a(n) = 2^(n-3)*binomial(n+3, 3)*(n+8). - _Brad Clardy_, Mar 08 2012 [See a comment in A053120 on subdiagonals. - _Wolfdieter Lang_, Jan 03 2020]

%F E.g.f.: (1/3)*exp(2*x)*(3 + 21*x + 27*x^2 + 10*x^3 + x^4). - _Stefano Spezia_, Aug 17 2019

%p a := n->n*(n+1)*(n+2)*(n+7)*2^(n-5)/3;

%o (MAGMA) [2^(n-1)/4*Binomial(n+3,3)*(n+8) : n in [0..25]]; // _Brad Clardy_, Mar 08 2012

%Y Cf. A039991 (see column 8), A003472 (partial sums), A053120.

%K nonn,easy

%O 0,2

%A _Simon Plouffe_

%E Name clarified by _Wolfdieter Lang_, Nov 26 2019

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Last modified March 6 20:37 EST 2021. Contains 341850 sequences. (Running on oeis4.)