More details on the table A101024/A101025: Explicit form of the rational polynomials R(n,x):=hypergeom([-n,-n-1],[1/2],x/2): R(n,x) = 1 + sum((binomial(n,m)*binomial(n+1,m)/binomial(2*m,m))*(2*x)^m,m=1..n), n>=0. The rational polynomials R(n,x):=hypergeom([-n,-n-1],[1/2],x/2) = sum(r(n,m)*x^m ,m=0..n), are, for n=0..10: n 0 1 1 1+2*x 2 1+6*x+2*x^2 3 1+12*x+12*x^2+8/5*x^3 4 1+20*x+40*x^2+16*x^3+8/7*x^4 5 1+30*x+100*x^2+80*x^3+120/7*x^4+16/21*x^5 6 1+42*x+210*x^2+280*x^3+120*x^4+16*x^5+16/33*x^6 7 1+56*x+392*x^2+784*x^3+560*x^4+448/3*x^5+448/33*x^6+128/429*x^7 8 1+72*x+672*x^2+9408/5*x^3+2016*x^4+896*x^5+1792/11*x^6+1536/143*x^7+128/715*x^8 9 1+90*x+1080*x^2+4032*x^3+6048*x^4+4032*x^5+13440/11*x^6+23040/143*x^7+1152/143*x^8+256/2431*x^9 10 1+110*x+1650*x^2+7920*x^3+15840*x^4+14784*x^5+6720*x^6+19200/13*x^7+192013*x^8+1280/221*x^9+256/4199*x^10 The rational coefficients furnish the triangle r(n,m), m=0..n, for n=0..10: n 0 1 1 1 2 2 1 6 2 3 1 12 12 8/5 4 1 20 40 16 8/7 5 1 30 100 80 120/7 16/21 6 1 42 210 280 120 16 16/33 7 1 56 392 784 560 448 448/33 128/429 8 1 72 672 9408/5 2016 896 1792/11 1536/143 128/715 9 1 90 1080 4032 6048 4032 13449/11 23040/143 1152/143 256/2431 10 1 110 1650 7920 15840 14784 6720 19200/13 192013 1280/221 256/4199 Written as rational sequence (for rows n=0..10): [1, 1, 2, 1, 6, 2, 1, 12, 12, 8/5, 1, 20, 40, 16, 8/7, 1, 30, 100, 80, 120/7, 16/21, 1, 42, 210, 280, 120, 16, 16/33,\ 1, 56, 392, 784, 560, 448/3, 448/33, 128/429, 1, 72, 672, 9408/5, 2016, 896, 1792/11, 1536/143, 128/715,\ 1, 90, 1080, 4032, 6048, 4032, 13440/11, 23040/143, 1152/143, 256/2431,\ 1, 110, 1650, 7920, 15840, 14784, 6720, 19200/13, 1920/13, 1280/221, 256/4199] The numerator sequence gives triangle A101024(n): a(n,m) tabl head (triangle) for A101022 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 2 0 0 0 0 0 0 0 0 2 1 6 2 0 0 0 0 0 0 0 3 1 12 12 8 0 0 0 0 0 0 4 1 20 40 16 8 0 0 0 0 0 5 1 30 100 80 120 16 0 0 0 0 6 1 42 210 280 120 16 16 0 0 0 7 1 56 392 784 560 448 448 128 0 0 8 1 72 672 9408 2016 896 1792 1536 128 0 9 1 90 1080 4032 6048 4032 13440 23040 1152 256 . . . The denominator sequence gives triangle A101025(n): a(n,m) tabl head (triangle) for A101024 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 5 0 0 0 0 0 0 4 1 1 1 1 7 0 0 0 0 0 5 1 1 1 1 7 21 0 0 0 0 6 1 1 1 1 1 1 33 0 0 0 7 1 1 1 1 1 3 33 429 0 0 8 1 1 1 5 1 1 11 143 715 0 9 1 1 1 1 1 1 11 143 143 2431 . . . #######################################################################################################################################