Assume f(n)=a0+a1n+a2n2+a3n3+a4n4 where 0≤n≤4.
f(0)=a0=4;
f(1)=a0+a1+a2+a3+a4=30, a1+a2+a3+a4=26;
f(2)=a0+2a1+4a2+8a3+16a4=120, 2a1+4a2+8a3+16a4=116;
f(3)=a0+3a1+9a2+27a3+81a4=340, 3a1+9a2+27a3+81a4=336;
f(4)=a0+4a1+16a2+64a3+256a4=780, 4a1+16a2+64a3+256a4=776;
(1) f(2)-2f(1)=2a2+6a3+14a4=64, a2+3a3+7a4=32;
(2) f(3)-3f(1)=6a2+24a3+78a4=258, a2+4a3+13a4=43;
(3) f(4)-4f(1)=12a2+60a3+252a4=672, a2+5a3+21a4=56;
(4) (2)-(1)=a3+6a4=11;
(5) (3)-(2)=a3+8a4=13;
(5)-(4)=2a4=2, a4=1; a3=5, a2=32-15-7=10, a1=26-10-5-1=10
f(n)=4+10n+10n2+5n3+n4 for n≥0.
f(n)=4+10(n-1)+10(n-1)2+5(n-1)3+(n-1)4 for n≥1.
Using this formula we get the series: 4, 30, 120, 340, 780, 1554, 2800, 4680, ...