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A generalization of unsigned Lah numbers, called L[3,2].","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:           5,           1;\n   3:          42,          16,           1;\n   4:         490,         270,          33,          1;\n   5:        7280,        5200,         936,         56,         1;\n   6:      131040,      114400,       27040,       2400,        85,        1;\n   7:     2766400,     2845440,      829920,      97280,      5130,      120,       1;\n   8:    66946880,    79119040,    27387360,    3979360,    277970,     9702,     161,     1;\n   9:  1825824000,  2434432000,   973772800,  168537600,  14630000,   677600,   16800,   208,   1;\n  10: 55383328000, 82162080000, 37246809600, 7471833600, 774219600, 44629200, 1470000, 27216, 261, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2-2*x)*P(n,x) = (3*n^2+7*n-6-(5*n-4)*x-2*x^2)*P(n-1,x) + (3*n*x-6*x^2)*d/dx 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[((3*n^2 + 7*n - 6 - (5*n - 4)*x - 2*x^2)*P[n - 1, x] + (3*n*x - 6*x^2)*D[P[n - 1, x], x])/(n + 2 - 2*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] = ((3*n^2 + 7*n - 6 - (5*n - 4)*x - 2*x^2)*v[i - 1] + (3*n*x - 6*x^2)*deriv(v[i - 1], x))/(n + 2 - 2*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-e18d516553291356","steps":["D+1"],"edges":["e-c22cc344febafff9"],"text":"A290598 \u2192 D+1"},"headline":"A290598 \u2192 D+1","chain":null}