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 A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k. 3
 1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Rising and falling diagonals are A008999, A124400. Subtriangle of triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2012 A102756 is jointly generated with A209130 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Mar 05 2012 LINKS FORMULA Sum_{k, 0<=k<=n}x^k*T(n,k) = A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for n= -1, 0, 1, 2, 3 respectively. Sum_{k, 0<=k<=n}(-2)^k*T(n,k) = [1,0,3,0,9,0,27,0,81,0,...]for n=0,1,2,3,4,5,..., powers of 3 alternating with zeros. T(n,n-1) = 2*A001629(n+1) for n>=1. T(n,n) = Fibonacci(n+1) = A000045(n+1). T(n,0) = n+1. T(n,1) = A000292(n) for n>=1. T(n+1,2) = binomial(n+4,n-1)+binomial(n+2,n-1)= A051747(n) for n>=1. G.f.: 1/(1-(2+y)*x+(1+y)*(1-y)*x^2). - Philippe Deléham, Feb 17 2012 EXAMPLE Triangle begins: 1; 2, 1; 3, 4, 2; 4, 10, 10, 3; 5, 20, 31, 20, 5; 6, 35, 76, 78, 40, 8; 7, 56, 161, 232, 184, 76, 13; 8, 84, 308, 582, 636, 406, 142, 21; 9, 120, 546, 1296, 1831, 1604, 861, 260, 34; 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55; Triangle (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins: 1 1, 0 2, 1, 0 3, 4, 2, 0 4, 10, 10, 3, 0 5, 20, 31, 20, 5, 0 6, 35, 76, 78, 40, 8, 0 7, 56, 161, 232, 184, 76, 13, 0 MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]; v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%]    (* A102756 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A209130 *) (* Clark Kimberling, Mar 05 2012 *) CROSSREFS Cf. A209130. Sequence in context: A120058 A208532 A245334 * A086614 A108959 A208750 Adjacent sequences:  A102753 A102754 A102755 * A102757 A102758 A102759 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Dec 18 2006 STATUS approved

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Last modified October 23 08:57 EDT 2019. Contains 328345 sequences. (Running on oeis4.)