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

1, 1111, 1122211, 1123333211, 1123445443211, 1123456666543211, 1123457788877543211, 1123457901110987543211, 1123457912334332087543211, 1123457913456666543087543211, 1123457913568899988653087543211, 1123457913580123333209753087543211, 1123457913581245667665420753087543211

Offset: 3

Author

Ilya Gutkovskiy, Aug 13 2016

Keywords

• nonn
• easy

Comments

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).

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

The first seven members are palindromes.

Links

• Index entries related to Gaussian binomial coefficients
• Index entries for linear recurrences with constant coefficients, signature (1111,-112110,1111000,-1000000)

Crossrefs

Cf. A002275, A022174, A109242, A147816.

Programs

Mathematica

Table[((10^n - 100) (10^n - 10) (10^n - 1))/890109000, {n, 0, 15}]
Table[QBinomial[n, 3, 10], {n, 3, 15}]

Formula

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

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

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

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

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