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A275944 Gaussian binomial coefficient [n,3] for q = 10. 0

%I

%S 1,1111,1122211,1123333211,1123445443211,1123456666543211,

%T 1123457788877543211,1123457901110987543211,1123457912334332087543211,

%U 1123457913456666543087543211,1123457913568899988653087543211,1123457913580123333209753087543211,1123457913581245667665420753087543211

%N Gaussian binomial coefficient [n,3] for q = 10.

%C More generally, the ordinary generation function for the Gaussian binomial coefficients [n,k]_q is x^k/Product_{m=0..k} (1 - q^m*x).

%C Convolution of A002275 and A147816 (considering offset: 0, 0, 1, 1100, 1110000, ...).

%C The first seven members are palindromes.

%H <a href="/index/Ga#Gaussian_binomial_coefficients">Index entries related to Gaussian binomial coefficients</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1111,-112110,1111000,-1000000)

%F O.g.f.: x^3/((1 - x)*(1 - 10*x)*(1 - 100*x)*(1 - 1000*x)).

%F E.g.f.: (-1000 + 1110*exp(9*x) - 111*exp(99*x) + exp(999*x))*exp(x)/890109000.

%F a(n) = 1111*a(n-1) - 112110*a(n-2) + 1111000*a(n-3) - 1000000*a(n-4).

%F a(n) = ((10^n - 100)*(10^n - 10)*(10^n - 1))/890109000.

%F a(n) = Product_{i=0..2} (1 - 10^(n-i))/(1 - 10^(i+1)).

%t Table[((10^n - 100) (10^n - 10) (10^n - 1))/890109000, {n, 0, 15}]

%t Table[QBinomial[n, 3, 10], {n, 3, 15}]

%Y Cf. A002275, A022174, A109242, A147816.

%K nonn,easy

%O 3,2

%A _Ilya Gutkovskiy_, Aug 13 2016

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Last modified February 18 23:26 EST 2018. Contains 299330 sequences. (Running on oeis4.)