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0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   11,      44,        36;\n   2:   26,     210,       420,        240;\n   3:   57,     832,      3060,       4080,        1800;\n   4:  120,    2982,     17976,      42000,       42000,        15120;\n   5:  247,   10068,     93492,     340704,      579600,       463680,       141120;\n   6:  502,   32730,    450540,    2404080,     6189120,      8255520,      5503680,      1451520;\n   7: 1013,  103784,   2063700,   15500880,    56662200,    112069440,    122774400,     70156800,     16329600;\n   8: 2036,  323598,   9125952,   93944400,   468507600,   1288813680,   2063033280,   1916006400,    958003200,    199584000;\n   9: 4083,  997372,  39355668,  544703808,  3606451200,  13254278400,  29073401280,  39017341440,  31434480000,  13970880000,   2634508800;\n  10: 8178, 3049410, 166644660, 3057054000, 26348562240, 125855009280, 363312069120, 660686906880, 762498696960, 542269728000, 216907891200, 37362124800;\n","recurrence_sentence":"The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = -(2*n+4+(n^2+13*n+22)*x+(5*n^2+35*n+50)*x^2+(9*n^2+47*n+58)*x^3+(7*n^2+31*n+34)*x^4+(2*n^2+8*n+8)*x^5)*P(n-2,x) - ((n+2)*x+(n^2+8*n+12)*x^2+(4*n^2+22*n+28)*x^3+(6*n^2+28*n+32)*x^4+(4*n^2+17*n+18)*x^5+(n^2+4*n+4)*x^6)*d/dx P(n-2,x) + (3*n+4+(n^2+15*n+18)*x+(3*n^2+27*n+30)*x^2+(3*n^2+21*n+22)*x^3+(n^2+6*n+6)*x^4)*P(n-1,x) + ((n+1)*x+(4*n+4)*x^2+(6*n+6)*x^3+(4*n+4)*x^4+(n+1)*x^5)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 11 + 44*x + 36*x^2, P(2,x) = 26 + 210*x + 420*x^2 + 240*x^3.","mathematica":"P[1, x_] := 11 + 44*x + 36*x^2;\nP[2, x_] := 26 + 210*x + 420*x^2 + 240*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (n^2 + 13*n + 22)*x + (5*n^2 + 35*n + 50)*x^2 + (9*n^2 + 47*n + 58)*x^3 + (7*n^2 + 31*n + 34)*x^4 + (2*n^2 + 8*n + 8)*x^5)*P[n - 2, x] - ((n + 2)*x + (n^2 + 8*n + 12)*x^2 + (4*n^2 + 22*n + 28)*x^3 + (6*n^2 + 28*n + 32)*x^4 + (4*n^2 + 17*n + 18)*x^5 + (n^2 + 4*n + 4)*x^6)*D[P[n - 2, x], x] + (3*n + 4 + (n^2 + 15*n + 18)*x + (3*n^2 + 27*n + 30)*x^2 + (3*n^2 + 21*n + 22)*x^3 + (n^2 + 6*n + 6)*x^4)*P[n - 1, x] + ((n + 1)*x + (4*n + 4)*x^2 + (6*n + 6)*x^3 + (4*n + 4)*x^4 + (n + 1)*x^5)*D[P[n - 1, x], x])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 11 + 44*x + 36*x^2; v[2] = 26 + 210*x + 420*x^2 + 240*x^3; for(n = 3, N, my(i = n); v[i] = (-(2*n + 4 + (n^2 + 13*n + 22)*x + (5*n^2 + 35*n + 50)*x^2 + (9*n^2 + 47*n + 58)*x^3 + (7*n^2 + 31*n + 34)*x^4 + (2*n^2 + 8*n + 8)*x^5)*v[i - 2] - ((n + 2)*x + (n^2 + 8*n + 12)*x^2 + (4*n^2 + 22*n + 28)*x^3 + (6*n^2 + 28*n + 32)*x^4 + (4*n^2 + 17*n + 18)*x^5 + (n^2 + 4*n + 4)*x^6)*deriv(v[i - 2], x) + (3*n + 4 + (n^2 + 15*n + 18)*x + (3*n^2 + 27*n + 30)*x^2 + (3*n^2 + 21*n + 22)*x^3 + (n^2 + 6*n + 6)*x^4)*v[i - 1] + ((n + 1)*x + (4*n + 4)*x^2 + (6*n + 6)*x^3 + (4*n + 4)*x^4 + (n + 1)*x^5)*deriv(v[i - 1], x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-41809911e05bb4ba","steps":["D (shift +2)","x \u21a6 x/(x+1)"],"edges":["e-2ca40a069e1a39d6","e-ed9846954eb8ab72"],"text":"A257609 \u2192 D (shift +2) \u2192 x \u21a6 x/(x+1)"},"headline":"A257609 \u2192 D (shift +2) \u2192 x \u21a6 x/(x+1)","chain":null}