Here’s how I solve this sort of problem.
First step: subtract the first term from each of the other values to produce a series with one fewer term: 19, 33, 38, 42, 45 and call these algebraically T₁ to T₅.
From these we are going to create 5 equations each containing 5 unknowns and we are basing them on the polynomial:
Tᵣ=a₅r⁵+a₄r⁴+a₃r³+a₂r²+a₁r, the coefficients being the unknowns to be found.
Second step: the primary equations:
① a₅+a₄+a₃+a₂+a₁=19 for r=1
② 32a₅+16a₄+8a₃+4a₂+2a₁=33 for r=2
③ 243a₅+81a₄+27a₃+9a₂+3a₁=38 for r=3
④ 1024a₅+256a₄+64a₃+16a₂+4a₁=42 for r=4
⑤ 3125a₅+625a₄+125a₃+25a₂+5a₁=45 for r=5
Third step: first reduction:
Subtract multiples of ① from each of the other equations using the rule Equation r minus r times Equation 1:
⑥ 30a₅+14a₄+6a₃+2a₂=-5
⑦ 240a₅+78a₄+24a₃+6a₂=-19
⑧ 1020a₅+252a₄+60a₃+12a₂=-34
➈ 3120a₅+620a₄+120a₃+20a₂=-50, which can be reduced to:
➈ 312a₅+62a₄+12a₃+2a₂=-5
Fourth step: second reduction:
Inspect the 4 equations above and manipulate them to eliminate another variable coefficient.
⑨-⑥: 282a₅+48a₄+6a₃=0
⑧-2⑦: 540a₅+96a₄+12a₃=4
Fifth step: third reduction:
We can see that doubling the first of these two equations gives us:
564a₅+96a₄+12a₃=0 and if we now subtract the second equation from this we get: 24a₅=-4, so a₅=-⅙.
Sixth step: substitution:
282a₅+48a₄+6a₃=0, -47+48a₄+6a₃=0, 48a₄+6a₃=47.
540a₅+96a₄+12a₃=4, -90+96a₄+12a₃=4, 96a₄+12a₃=94, which is the same as above and confirms the process.
Now continue the substitution of a₅=-⅙.
⑥ 30a₅+14a₄+6a₃+2a₂=-5⇒14a₄+6a₃+2a₂=0⇒7a₄+3a₃+a₂=0
⑦ 240a₅+78a₄+24a₃+6a₂=-19⇒78a₄+24a₃+6a₂=21⇒26a₄+8a₃+2a₂=7
⑧ 1020a₅+252a₄+60a₃+12a₂=-34⇒252a₄+60a₃+12a₂=136⇒63a₄+15a₃+3a₂=34
➈ 312a₅+62a₄+12a₃+2a₂=-5⇒62a₄+12a₃+2a₂=47
Seventh step: elimination of more unknowns:
⑦-⑥: 12a₄+2a₃=7
⑨-⑥: 48a₄+6a₃=47
⑨-⑥-3(⑦-⑥): 12a₄=26, a₄=13/6, a₃=½(7-26)=-19/2.
Eighth step: substitute all discovered coefficients: a₅, a₄, a₃.
a₂=-7(13/6)-3(-19/2)=40/3.
a₁=19-(-1/6+13/6-19/2+40/3)=79/6.
Tᵣ=-r⁵/6+13r⁴/6-19r³/2+40r²/3+79r/6.
Tᵣ+11=-r⁵/6+13r⁴/6-19r³/2+40r²/3+79r/6+11 produces the series:
11, 30, 44, 49, 53, 56, 30, -101, -481, -1342, -3024, -5995.
Please note that there will probably be other formulas generating a series with the given 6 terms.