Triangular numbers in Mathematica

Some functions for triangular numbers in Mathematica, sorted by byte count

15: ((2#+1)^2-1)/8&
17: Tr[Range@#^3]^.5&
17: ⌊(2#+1)^2/8⌋&
18: ((#+1)^3-#^3-1)/6&
19: Sqrt@Tr[Range@#^3]&
19: -StirlingS1[#+1,#]&
19: Sum[1,{i,#},{j,i}]&
20: f@0=0;f@i_:=i+f[i-1]
20: Tr[{#+1,-#,-1}^3/6]&
22: ⌊(#+1)^3/(#+2)⌋/2&
23: Tr@Table[i^3,{i,#}]^.5&
26: Sqrt@Sum[i*j,{i,#},{j,#}]&
27: Sum[i^2(-1)^i,{i,#}](-1)^#&
28: D[x/(1-x)^3,{x,#}]/#!/.x->0&
28: (For[i=n=0,i<#,n+=i,i++];n)&
29: n=0;(Do[n+=i,{i,#}];Print@n)&
29: EdgeCount@CompleteGraph[#+1]&
29: Length@Expand[(x+y+z)^(#-1)]&	(note: doesn't work for 1)
29: Coefficient[(1+x)^(#+1),x^2]&
29: ⌊(#+1)/(Exp[2/(#+1)]-1)⌋&
30: Nest[#+(1+Sqrt[1+8#])/2&,0,#]&
31: Sum[EulerPhi@k⌊#/k⌋,{k,#}]&
33: Series[x/(1-x)^3,{x,0,#}][[3,#]]&
34: Boole[#2>=#]&~Array~{#,#}~Total~2&	(thanks to Mark S., https://codegolf.stackexchange.com/questions/133109/sum-of-all-integers-from-1-to-n/133160#comment339998_133160)
37: SeriesCoefficient[x/(1-x)^3,{x,0,#}]&
37: Det@Array[Binomial[1+##,#]&,{#,#}-1]&	(note: doesn't work for 1)
39: Length@Solve[0<a<=b<=#,{a,b},Integers]&
40: UpperTriangularize@Table[1,#,#]~Total~2&
42: Last@LinearRecurrence[{3,-3,1},{1,3,6},#]&
42: CellularAutomaton[50,{{1},0},#-1]~Total~2&
45: 1/Integrate[4Sin@x^(2#-1)Cos@x^3,{x,0,Pi/2}]&