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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:          43,           16,           1;\n   4:         521,          273,          33,          1;\n   5:        8113,         5360,         942,         56,         1;\n   6:      154001,       120965,       27530,       2410,        85,        1;\n   7:     3442915,      3101376,      858435,      98440,      5145,      120,       1;\n   8:    88491173,     89229301,    28881951,    4070255,    280315,     9723,     161,     1;\n   9:  2568458401,   2849432608,  1050084028,  174704656,  14868070,   681856,   16828,   208,   1;\n  10: 83033591453, 100050806025, 41174715492, 7867277796, 794523366, 45172638, 1477140, 27252, 261, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + (18+6*x)*d/dx P(n-1,x) + (9+9*x)*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[(5 + x)*P[n - 1, x] + (18 + 6*x)*D[P[n - 1, x], x] + (9 + 9*x)*D[P[n - 1, x], {x, 2}]];\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] = (5 + x)*v[i - 1] + (18 + 6*x)*deriv(v[i - 1], x) + (9 + 9*x)*deriv(deriv(v[i - 1], x), 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":["x \u21a6 x+1"],"edges":["e-fd137cc1379ca530"],"text":"A290598 \u2192 x \u21a6 x+1"},"headline":"A290598 \u2192 x \u21a6 x+1","chain":null}