I've been trying to learn generating functions and I got stuck finding the generating function for the sequence $1, 2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4$. I was able to notice that each term of the sequence can be counted as $\binom{n}{3}$ such that there are 1 one's, 3 two's, 6 three's and 10 four's. I've tried using differencing and I got that: $$(1-x)A = 1 + x +x^4 +x^{10}$$Could anyone help me find the closed form? Thank you for your time!
2026-04-12 21:47:50.1776030470
Finding generating function for a sequence associated wth binomial coefficients
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If the sequence goes on with 15 fives then the generating function should be $$f(x)=\sum_{n=1}^{\infty}n\sum_{k=\binom{n+1}{3}}^{\binom{n+2}{3}-1}x^k=\frac{\sum_{n=0}^{\infty}x^{\binom{n+2}{3}}}{1-x}.$$ where at the numerator we have the g.f. of the characteristic function of tetrahedral numbers. I don't think there is a simple closed formula for that.