A124266 Variant sequence generated by solving the order n X n linear problem [H]x = b where b is the unit vector and the sequence term is given by the denominator of the last unknown xn.
1, 1, 1, 3, 6, 10, 150, 525, 980, 24696, 740880, 2910600, 82328400, 168185160, 1870592724
Offset: 1
Programs
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Maxima
HilbertWarren(fun, order) := ( Unity[i,j] := 1, A : genmatrix(fun, order, order), B : genmatrix(Unity, 1, order), App : invert(triangularize(A)), Xp : App . B, 1/Xp[order] ); findWarrenSequenceTerms(fun, a, b) := ( L : append(), for order: a next order+1 thru b do L: cons(first(HilbertWarren(fun,order)), L), S : reverse(L) ); k : 15; hilbert[i,j] := 1/(i + j - 1); findWarrenSequenceTerms(hilbert, 1, k); hilbertA0[i,j] := (i + j + 0)/(i + j - 1); /* sum 1 */ findWarrenSequenceTerms(hilbertA0, 1, k); hilbertA1[i,j] := (i + j + 1)/(i + j - 1); /* sum 2: there are lots of these, increment numerator */ findWarrenSequenceTerms(hilbertA1, 1, k); hilbertD1[i,j] := (i - j + 1)/(i + j - 1); /* difference 1 */ findWarrenSequenceTerms(hilbertD1, 1, k); hilbertP1[i,j] := (i * j + 0)/(i + j - 1); /* product 1 */ findWarrenSequenceTerms(hilbertP1, 1, k); hilbertQ1[i,j] := (i / j)/(i + j - 1); /* quotient 1 */ findWarrenSequenceTerms(hilbertQ1, 1, k); /* Program does not display the correct sequence. Robert C. Lyons, Jul 30 2025 */
Formula
[H] is defined by hilbertWarrenA1[i,j]:=(1-j+i)/(-1+j+i) where numbering starts at 1.