{"schema":"vela.problem-packet.v0.1","problem":849,"statement":"Is it true that, for every integer $t\\geq 1$, there is some integer $a$ such that\\[\\binom{n}{k}=a\\](with $1\\leq k\\leq n/2$) has exactly $t$ solutions?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A003015","name":"Numbers that occur 5 or more times in Pascal's triangle.","terms":"1,120,210,1540,3003,7140,11628,24310,61218182743304701891431482520","url":"https://oeis.org/A003015"},{"id":"A003016","name":"Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).","terms":"0,3,1,2,2,2,3,2,2,2,4,2,2,2,2,4,2,2,2,2,3,4,2,2,2,2,2,2,4,2,2,2,2,2,2,4,4,2,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,2,2,4,4,2,2,2,","url":"https://oeis.org/A003016"},{"id":"A059233","name":"Number of rows in which n appears in Pascal's triangle A007318.","terms":"1,1,1,1,2,1,1,1,2,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,","url":"https://oeis.org/A059233"},{"id":"A090162","name":"Values of binomial(Fibonacci(2k)*Fibonacci(2k+1),Fibonacci(2k-1)*Fibonacci(2k)-1).","terms":"1,3003,61218182743304701891431482520","url":"https://oeis.org/A090162"},{"id":"A098565","name":"Numbers that appear as binomial coefficients exactly 6 times.","terms":"120,210,1540,7140,11628,24310,61218182743304701891431482520","url":"https://oeis.org/A098565"},{"id":"A180058","name":"Smallest number occurring in exactly n rows of Pascal's triangle.","terms":"2,6,120,3003","url":"https://oeis.org/A180058"},{"id":"A182237","name":"Numbers occurring exactly in 2 rows of Pascal's triangle.","terms":"6,10,15,20,21,28,35,36,45,55,56,66,70,78,84,91,105,126,136,153,165,171,190,220,231,252,253,276,286,300,325,330,351,364,3","url":"https://oeis.org/A182237"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}