More details on the table A101026/A101027: Explicit form of the rational polynomials R(n,x):=hypergeom([-n,-n-2],[1/2],x/2): R(n,x) = 1 + sum((binomial(n,m)*binomial(n+2,m)/binomial(2*m,m))*(2*x)^m,m=1..n),n>=0. The rational polynomials R(n,x):=hypergeom([-n,-n-2],[1/2],x/2) = sum(r(n,m)*x^m ,m=0..n), are, for n=0..10: n 0 1 1 1+3*x 2 1+8*x+4*x^2 3 1+15*x+20*x^2+4*x^3 4 1+24*x+60*x^2+32*x^3+24/7*x^4 5 1+35*x+140*x^2+140*x^3+40*x^4+8/3*x^5 6 1+48*x+280*x^2+448*x^3+240*x^4+128/3*x^5+64/33*x^6 7 1+63*x+504*x^2+1176*x^3+1008*x^4+336*x^5+448/11*x^6+192/143*x^7 8 1+80*x+840*x^2+2688*x^3+3360*x^4+1792*x^5+4480/11*x^6+5120/143*x^7+128/143*x^8 9 1+99*x+1320*x^2+5544*x^3+9504*x^4+7392*x^5+2688*x^6+5760/13*x^7+384/13*x^8+128/221*x^9 10 1+120*x+1980*x^2+10560*x^3+23760*x^4+25344*x^5+13440*x^6+46080/13*x^7+5760/13*x^8+5120/221*x^9+1536/4199*x^10 The rational coefficients furnish the triangle r(n,m), m=0..n, for n=0..10: n 0 1 1 1 3 2 1 8 4 3 1 15 20 4 4 1 24 60 32 24/7 5 1 35 140 140 40 8/3 6 1 48 280 448 240 128/3 64/33 7 1 63 504 1176 1008 336 448/11 192/143 8 1 80 840 2688 3360 1792 4480/11 5120/143 128/143 9 1 99 1320 5544 9504 7392 2688 5760/13 384/13 128/221 10 1 120 1980 10560 23760 25344 13440 46080/13 5760/13 5120/221 1536/4199 Written as rational sequence (for rows n=0..10): [1, 1, 3, 1, 8, 4, 1, 15, 20, 4, 1, 24, 60, 32, 24/7, 1, 35, 140, 140, 40, 8/3, \ 1, 48, 280, 448, 240, 128/3, 64/33, 1, 63, 504, 1176, 1008, 336, 448/11, 192/143, \ 1, 80, 840, 2688, 3360, 1792, 4480/11, 5120/143, 128/143, \ 1, 99, 1320, 5544, 9504, 7392, 2688, 5760/13, 384/13, 128/221, \ 1, 120, 1980, 10560, 23760, 25344, 13440, 46080/13, 5760/13, 5120/221, 1536/4199] The numerator sequence gives triangle A101026(n): a(n,m) tabl head (triangle) for A101026 n\m 0 1 2 3 4 5 6 7 8 9 . . . 0 1 0 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 0 2 1 8 4 0 0 0 0 0 0 0 3 1 15 20 4 0 0 0 0 0 0 4 1 24 60 32 24 0 0 0 0 0 5 1 35 140 140 40 8 0 0 0 0 6 1 48 280 448 240 128 64 0 0 0 7 1 63 504 1176 1008 336 448 192 0 0 8 1 80 840 2688 3360 1792 4480 5120 128 0 9 1 99 1320 5544 9504 7392 2688 5760 384 128 . . . The denominator sequence gives triangle A101027(n): a(n,m) tabl head (triangle) for A101027 n\m 0 1 2 3 4 5 6 7 8 9 . . . 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 2 1 1 1 0 0 0 0 0 0 0 3 1 1 1 1 0 0 0 0 0 0 4 1 1 1 1 7 0 0 0 0 0 5 1 1 1 1 1 3 0 0 0 0 6 1 1 1 1 1 3 33 0 0 0 7 1 1 1 1 1 1 11 143 0 0 8 1 1 1 1 1 1 11 143 143 0 9 1 1 1 1 1 1 1 13 13 221 . . . #######################################################################################################################################