A054331 - OEIS

%I #21 Feb 02 2023 18:28:56

%S 1,60,1584,27456,366080,4073472,39690240,349274112,2835283968,

%T 21554790400,155194490880,1067345510400,7058711642112,45127489814528,

%U 280101660917760,1693862087098368,10009185060126720,57935518230380544

%N One eighth of eighth unsigned column of Lanczos' triangle A053125.

%D C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

%H G. C. Greubel, <a href="/A054331/b054331.txt">Table of n, a(n) for n = 0..1000</a>

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

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (32,-448,3584,-17920,57344,-114688,131072,-65536).

%F a(n) = 2^(2*n-3)*binomial(2*n+8, 7) = -A053125(n+7, 7)/8 = A054326(n)/8.

%F G.f. (1+4*x)*(1+24*x+16*x^2)/(1-4*x)^8.

%t Table[4^n Binomial[2n+8,7]/8,{n,0,20}] (* _Harvey P. Dale_, Nov 03 2011 *)

%t LinearRecurrence[{32,-448,3584,-17920,57344,-114688,131072,-65536},{1,60,1584,27456,366080,4073472,39690240,349274112},20] (* _Harvey P. Dale_, Feb 25 2022 *)

%o (PARI) vector(20, n, n--; 2^(2*n-3)*binomial(2*n+8, 7)) \\ _G. C. Greubel_, Jul 22 2019

%o (Magma) [2^(2*n-3)*Binomial(2*n+8, 7): n in [0..20]]; // _G. C. Greubel_, Jul 22 2019

%o (Sage) [2^(2*n-3)*binomial(2*n+8, 7) for n in (0..20)] # _G. C. Greubel_, Jul 22 2019

%o (GAP) List([0..20], n-> 2^(2*n-3)*Binomial(2*n+8, 7)); # _G. C. Greubel_, Jul 22 2019

%Y Cf. A054326, A053125.

%K easy,nice,nonn

%O 0,2

%A _Wolfdieter Lang_