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 A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros. 5
 1, 1, 1, 1, 2, 1, 2, 5, 4, 1, 1, 5, 8, 5, 1, 3, 13, 22, 18, 7, 1, 1, 9, 26, 35, 24, 8, 1, 4, 26, 70, 101, 84, 40, 10, 1, 1, 14, 61, 131, 160, 116, 49, 11, 1, 5, 45, 171, 363, 476, 400, 215, 71, 13, 1, 1, 20, 120, 363, 654, 752, 565, 275, 83, 14, 1, 6, 71, 356, 1017, 1856, 2282, 1932 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums form A099603, where A099603(n) = fibonacci(n+1)*2^[(n+1)/2]. Central coefficients of even-indexed rows form A082759, where A082759(n) = Sum_{k=0..n} binomial(n,k)*Trinomial(n,k). Antidiagonal sums form A099604. Matrix inverse equals triangle A104495, which is generated from self-convolutions of the Catalan sequence (A000108). LINKS FORMULA G.f.: (1 + (y+1)*x - (y+1)*x^2)/(1 - (y+1)*(y+2)*x^2 + (y+1)^2*x^4). EXAMPLE Rows begin: [1], [1,1], [1,2,1], [2,5,4,1], [1,5,8,5,1], [3,13,22,18,7,1], [1,9,26,35,24,8,1], [4,26,70,101,84,40,10,1], [1,14,61,131,160,116,49,11,1], [5,45,171,363,476,400,215,71,13,1], [1,20,120,363,654,752,565,275,83,14,1],... The binomial transform of row 2 equals column 2 of A027907: BINOMIAL[1,2,1] = [1,3,6,10,15,21,28,36,45,55,...]. The binomial transform of row 3 equals column 3 of A027907: BINOMIAL[2,5,4,1] = [2,7,16,30,50,77,112,156,210,...]. The binomial transform of row 4 equals column 4 of A027907: BINOMIAL[1,5,8,5,1] = [1,6,19,45,90,161,266,414,615,...]. The binomial transform of row 5 equals column 5 of A027907: BINOMIAL[3,13,22,18,7,1] = [3,16,51,126,266,504,882,1452,...]. PROG (PARI) {T(n, k)=polcoeff(polcoeff((1+(y+1)*x-(y+1)*x^2)/(1-(y+1)*(y+2)*x^2+(y+1)^2*x^4)+x*O(x^n), n, x)+y*O(y^k), k, y)} (PARI) {T(n, k)=(matrix(n+1, n+1, i, j, if(i>=j, polcoeff(polcoeff( (1+x*y/(1+x))/(1+x-y^2*(1-(1+4*x+O(x^i))^(1/2))^2/4+O(y^j)), i-1, x), j-1, y)))^-1)[n+1, k+1]} CROSSREFS Cf. A027907, A082759, A099603, A099604. Cf. A104495. Sequence in context: A318354 A348373 A106480 * A151703 A151691 A201780 Adjacent sequences: A099599 A099600 A099601 * A099603 A099604 A099605 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Oct 25 2004 STATUS approved

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Last modified February 3 04:25 EST 2023. Contains 360024 sequences. (Running on oeis4.)