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A generalization of unsigned Lah numbers, called L[3,1].","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 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:          1;\n   2:          2,           3;\n   3:         10,          30,           21;\n   4:         80,         360,          504,          225;\n   5:        880,        5280,        11088,         9900,          3213;\n   6:      12320,       92400,       258720,       346500,        224910,         57051;\n   7:     209440,     1884960,      6597360,     11781000,      11470410,       5819202,       1207953;\n   8:    4188800,    43982400,    184726080,    412335000,     535285800,     407344140,     169113420,     29652561;\n   9:   96342400,  1156108800,   5664933120,  15173928000,   24623146800,   24983773920,   15558434640,   5456071224,    827024985;\n  10: 2504902400, 33816182400, 189370621440, 591783192000, 1152363270240, 1461550774320, 1213557901920, 638360333208, 193523846490, 25810060563;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(6*n-14-(9*n^2-69*n+107)*x-(27*n^2-165*n+225)*x^2-(27*n^2-147*n+186)*x^3-(9*n^2-45*n+54)*x^4)*P(n-1,x) + (6-(18*n-84)*x-(72*n-270)*x^2-(108*n-366)*x^3-(72*n-228)*x^4-(18*n-54)*x^5)*d/dx P(n-1,x) + (9*x+45*x^2+90*x^3+90*x^4+45*x^5+9*x^6)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 14 - (9*n^2 - 69*n + 107)*x - (27*n^2 - 165*n + 225)*x^2 - (27*n^2 - 147*n + 186)*x^3 - (9*n^2 - 45*n + 54)*x^4)*P[n - 1, x] + (6 - (18*n - 84)*x - (72*n - 270)*x^2 - (108*n - 366)*x^3 - (72*n - 228)*x^4 - (18*n - 54)*x^5)*D[P[n - 1, x], x] + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*x^6)*D[P[n - 1, x], {x, 2}])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(6*n - 14 - (9*n^2 - 69*n + 107)*x - (27*n^2 - 165*n + 225)*x^2 - (27*n^2 - 147*n + 186)*x^3 - (9*n^2 - 45*n + 54)*x^4)*v[i - 1] + (6 - (18*n - 84)*x - (72*n - 270)*x^2 - (108*n - 366)*x^3 - (72*n - 228)*x^4 - (18*n - 54)*x^5)*deriv(v[i - 1], x) + (9*x + 45*x^2 + 90*x^3 + 90*x^4 + 45*x^5 + 9*x^6)*deriv(deriv(v[i - 1], x), x))/(1 + x)); 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-33e7d067844832dd","steps":["x \u21a6 x/(x+1)"],"edges":["e-3b4871ee267ab502"],"text":"A290596 \u2192 x \u21a6 x/(x+1)"},"headline":"A290596 \u2192 x \u21a6 x/(x+1)","chain":null}