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 A102051 Matrix inverse of triangle A101275 (number of SchrÃ¶der paths). 2
 1, -1, 1, 3, -4, 1, -9, 15, -7, 1, 31, -58, 36, -10, 1, -113, 229, -170, 66, -13, 1, 431, -924, 775, -372, 105, -16, 1, -1697, 3795, -3481, 1939, -691, 153, -19, 1, 6847, -15822, 15542, -9674, 4072, -1154, 210, -22, 1, -28161, 66801, -69276, 47012, -22446, 7606, -1788, 276, -25, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Row sums are {1,0,0,0...}. Absolute row sums form A006139. Column 0 forms signed A052709. Column 1 forms A102052. Column 2 forms A102053. LINKS FORMULA G.f.: 2/(1+y+(1-y)*sqrt(1+4*x-4*x^2)). T(n,m) = (-1)^(n-m)*(2*m+1)*Sum_{k=0..n} C(k,n-k)*C(2*k,k-m)/(m+k+1). - Vladimir Kruchinin, Apr 18 2015 EXAMPLE Rows begin: [1], [ -1,1], [3,-4,1], [ -9,15,-7,1], [31,-58,36,-10,1], [ -113,229,-170,66,-13,1], [431,-924,775,-372,105,-16,1], [ -1697,3795,-3481,1939,-691,153,-19,1], [6847,-15822,15542,-9674,4072,-1154,210,-22,1],... Matrix inverse equals triangle A101275: [1], [1,1], [1,4,1], [1,13,7,1], [1,44,34,10,1],... PROG (PARI) {T(n, k)=polcoeff(polcoeff(2/(2*y+(1-y)*(1+sqrt(1+4*x-4*x^2+x*O(x^n)))), n)+y*O(y^k), k)} (Maxima) T(n, m):=(-1)^(n-m)*(2*m+1)*(sum((binomial(k, n-k)*binomial(2*k, k-m))/(m+k+1), k, 0, n)); /* Vladimir Kruchinin, Apr 18 2015 */ CROSSREFS Cf. A101275, A006139, A052709, A102052, A102053, A039599. Sequence in context: A010611 A136158 A164948 * A078068 A054649 A138263 Adjacent sequences:  A102048 A102049 A102050 * A102052 A102053 A102054 KEYWORD sign,tabl AUTHOR Paul D. Hanna, Dec 27 2004 STATUS approved

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)