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 A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2*p-1)/2) = (2*p-1)*(column p of T), or [T^((2*p-1)/2)](m,0) = (2*p-1)*T(p+m,p+1) for all m>=1 and p>=0. 17
 1, 2, 1, 10, 4, 1, 74, 26, 6, 1, 706, 226, 50, 8, 1, 8162, 2426, 522, 82, 10, 1, 110410, 30826, 6498, 1010, 122, 12, 1, 1708394, 451586, 93666, 14458, 1738, 170, 14, 1, 29752066, 7489426, 1532970, 235466, 28226, 2754, 226, 16, 1, 576037442 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Column 0 is A000698 (related to double factorials), offset 1. Column 1 is A105616 (column 0 of T^(1/2), offset 1). The matrix logarithm divided by 2 yields the integer triangle A105629. Compare with triangular matrix A107717, which satisfies: SHIFT_LEFT(column 0 of A107717^((3*k-1)/3)) = (3*k-1)*(column k of A107717). LINKS FORMULA T(n, k) = 2*(k+1)*T(n, k+1) + Sum_{j=1..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>=0, with T(n, n) = 1 for n>=0. T(n, 0) = A000698(n+1) for n>=0. EXAMPLE SHIFT_LEFT(column 0 of T^(-1/2)) = -1*(column 0 of T); SHIFT_LEFT(column 0 of T^(1/2)) = 1*(column 1 of T); SHIFT_LEFT(column 0 of T^(3/2)) = 3*(column 2 of T); SHIFT_LEFT(column 0 of T^(5/2)) = 5*(column 3 of T). Triangle begins: 1; 2,1; 10,4,1; 74,26,6,1; 706,226,50,8,1; 8162,2426,522,82,10,1; 110410,30826,6498,1010,122,12,1; 1708394,451586,93666,14458,1738,170,14,1; 29752066,7489426,1532970,235466,28226,2754,226,16,1; ... Matrix square-root T^(1/2) is A105623 which begins: 1; 1,1; 4,2,1; 26,10,3,1; 226,74,19,4,1; 2426,706,167,31,5,1; ... compare column 0 of T^(1/2) to column 1 of T; also, column 1 of T^(1/2) equals column 0 of T. Matrix inverse square-root T^(-1/2) is A105620 which begins: 1; -1,1; -2,-2,1; -10,-4,-3,1; -74,-20,-7,-4,1; -706,-148,-39,-11,-5,1; ... compare column 0 of T^(-1/2) to column 0 of T. Matrix inverse T^-1 is A105619 which begins: 1; -2,1; -2,-4,1; -10,-2,-6,1; -74,-10,-2,-8,1; -706,-74,-10,-2,-10,1; -8162,-706,-74,-10,-2,-12,1; ... PROG (PARI) {T(n, k) = if(n=j, if(m==j, 1, if(m==j+1, -2*j, polcoeff(1/sum(i=0, m-j, (2*i)!/i!/2^i*x^i)+O(x^m), m-j)))))^-1)[n+1, k+1])} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) CROSSREFS Cf. A000698 (column 0), A105616 (column 1), A105617 (column 2), A105618 (row sums), A105619 (T^-1), A105620 (T^(-1/2)), A105623 (T^(1/2)), A105627 (T^(3/2)), A105629 (matrix log). Cf. A107717. Sequence in context: A202483 A110682 A110327 * A136216 A121334 A126450 Adjacent sequences:  A105612 A105613 A105614 * A105616 A105617 A105618 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Apr 16 2005 STATUS approved

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Last modified January 20 05:32 EST 2020. Contains 331067 sequences. (Running on oeis4.)