More details on the table A101022/A101023: Explicit form of the rational polynomials R(n,x):= hypergeom([1,1,1-n],[3/2,2],-x/2): R(n,x) = 1 + sum(binomial(n-1,m)/((m+1)*(2*m+1)*binomial(2*m,m))*(2*x)^m,m=1..n-1) The rational polynomials R(n,x):= hypergeom([1,1,1,-n],[3/2,2],-x/2) = sum(r(n,m)*x^m ,m=0..n-1), are, for n=1..11: n 1 1 2 1+1/6*x 3 1+1/3*x+2/45*x^2 4 1+1/2*x+2/15*x^2+1/70*x^3 5 1+2/3*x+4/15*x^2+2/35*x^3+8/1575*x^4 6 1+5/6*x+4/9*x^2+1/7*x^3+8/315*x^4+4/2079*x^5 7 1+x+2/3*x^2+2/7*x^3+8/105*x^4+8/693*x^5+16/21021*x^6 8 1+7/6*x+14/15*x^2+1/2*x^3+8/45*x^4+4/99*x^5+16/3003*x^6+2/6435*x^7 9 1+4/3*x+56/45*x^2+4/5*x^3+16/45*x^4+32/297*x^5+64/3003*x^6+16/6435*x^7+128/984555*x^8 10 1+3/2*x+8/5*x^2+6/5*x^3+16/25*x^4+8/33*x^5+64/1001*x^6+8/715*x^7+128/109395*x^8+64/1154725*x^9 11 1+5/3*x+2*x^2+12/7*x^3+16/15*x^4+16/33*x^5+160/1001*x^6+16/429*x^7+128/21879*x^8+128/230945*x^9+256/10669659*x^10 The rational coefficients furnish the triangle r(n,m), m=0..n-1, which for for n=1..11 is: n 1 1 2 1 1/6 3 1 1/3 2/45 4 1 1/2 2/15 1/70 5 1 2/3 4/15 2/35 8/1575 6 1 5/6 4/9 1/7 8/315 4/2079 7 1 1 2/3 2/7 8/105 8/693 16/21021 8 1 7/6 14/15 1/2 8/45 4/99 16/3003 2/6435 9 1 4/3 56/45 4/5 16/45 32/297 64/3003 16/6435 128/984555 10 1 3/2 8/5 6/5 16/25 8/33 64/1001 8/715 128/109395 64/1154725 11 1 5/3 2 12/7 16/15 16/33 160/1001 16/429 128/21879 128/230945 256/10669659 Written as rational sequence (n=1..11): [1, 1, 1/6, 1, 1/3, 2/45, 1, 1/2, 2/15, 1/70, 1, 2/3, 4/15, 2/35, 8/1575, 1, 5/6, 4/9, 1/7, 8/315, 4/2079, 1, 1, 2/3, 2/7, 8/105, 8/693, 16/21021, 1, 7/6, 14/15, 1/2, 8/45, 4/99, 16/3003, 2/6435, 1, 4/3, 56/45, 4/5, 16/45, 32/297, 64/3003, 16/6435, 128/984555, 1, 3/2, 8/5, 6/5, 16/25, 8/33, 64/1001, 8/715, 128/109395, 64/1154725, 1, 5/3, 2, 12/7, 16/15, 16/33, 160/1001, 16/429, 128/21879, 128/230945, 256/10669659] The numerator sequence gives table A101022(n,m): a(n,m) tabl head (triangle) for A101022 (with zeros above main diagonal) n\m 0 1 2 3 4 5 6 7 8 9 .... 1 1 0 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 3 1 1 2 0 0 0 0 0 0 0 4 1 1 2 1 0 0 0 0 0 0 5 1 2 4 2 8 0 0 0 0 0 6 1 5 4 1 8 4 0 0 0 0 7 1 1 2 2 8 8 16 0 0 0 8 1 7 14 1 8 4 16 2 0 0 9 1 4 56 4 16 32 64 16 128 0 10 1 3 8 6 16 8 64 8 128 64 . . . The denominator sequence gives table A101023(n,m): a(n,m) tabl head (triangle) for A101023 n\m 0 1 2 3 4 5 6 7 8 9 ... 1 1 0 0 0 0 0 0 0 0 0 2 1 6 0 0 0 0 0 0 0 0 3 1 3 45 0 0 0 0 0 0 0 4 1 2 15 70 0 0 0 0 0 0 5 1 3 15 35 1575 0 0 0 0 0 6 1 6 9 7 315 2079 0 0 0 0 7 1 1 3 7 105 693 21021 0 0 0 8 1 6 15 2 45 99 3003 6435 0 0 9 1 3 45 5 45 297 3003 6435 984555 0 10 1 2 5 5 25 33 1001 715 109395 1154725 . . . #######################################################################################################################################