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Coefficients of Chebyshev T polynomials: a(n) = A053120(n+12, n), n >= 0.
(Formerly M4907)
12

%I M4907 #54 Sep 26 2024 18:56:14

%S 1,13,98,560,2688,11424,44352,160512,549120,1793792,5637632,17145856,

%T 50692096,146227200,412778496,1143078912,3111714816,8341487616,

%U 22052208640,57567870976,148562247680,379364311040,959384125440

%N Coefficients of Chebyshev T polynomials: a(n) = A053120(n+12, n), n >= 0.

%C Binomial transform of A069039. - _Paul Barry_, Feb 19 2003

%C If X_1, X_2, ..., X_n are 2-blocks of a (2n+1)-set X then, for n >= 5, a(n-5) is the number of (n+6)-subsets of X intersecting each X_i, (i = 1, 2, ..., n). - _Milan Janjic_, Nov 18 2007

%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 G. C. Greubel, <a href="/A006976/b006976.txt">Table of n, a(n) for n = 0..1000</a>

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

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (14,-84,280,-560,672,-448,128).

%F G.f.: (1-x)/(1-2*x)^7.

%F a(n) = 2^n*binomial(n+5, 5)*(n+12)/12. [See a comment in A053120 on subdiagonal sequences. - _Wolfdieter Lang_, Jan 03 2020]

%F a(n) = Sum_{k = 0..floor((n+12)/2)} C(n+12,2*k)*C(k,6). - _Paul Barry_, May 15 2003

%F E.g.f.: (1/45)*exp(2*x)*(45 + 495*x + 1125*x^2 + 900*x^3 + 300*x^4 + 42*x^5 + 2*x^6). - _Stefano Spezia_, Jan 03 2020

%p seq(2^(n-1)*binomial(n+5,5)*(n+12)/6, n=0..25); # _G. C. Greubel_, Aug 27 2019

%t Table[2^(n-1)*Binomial[n+5,5]*(n+12)/6, {n,0,25}] (* _G. C. Greubel_, Aug 27 2019 *)

%t LinearRecurrence[{14,-84,280,-560,672,-448,128},{1,13,98,560,2688,11424,44352},30] (* _Harvey P. Dale_, Sep 26 2024 *)

%o (Magma) [2^(n-1)/6*Binomial(n+5,5)*(n+12) : n in [0..25]]; // _Brad Clardy_, Mar 10 2012

%o (PARI) vector(26, n, 2^(n-2)*binomial(n+4,5)*(n+11)/6) \\ _G. C. Greubel_, Aug 27 2019

%o (Sage) [2^(n-1)*binomial(n+5,5)*(n+12)/6 for n in (0..25)] # _G. C. Greubel_, Aug 27 2019

%o (GAP) List([0..25], n-> 2^(n-1)*Binomial(n+5,5)*(n+12)/6); # _G. C. Greubel_, Aug 27 2019

%Y a(n) = A039991(n+12, 12), A053120.

%Y Partial sums are in A002409.

%K nonn,easy

%O 0,2

%A _Simon Plouffe_

%E More terms from _James A. Sellers_, Aug 21 2000

%E Name clarified by _Wolfdieter Lang_, Nov 26 2019