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A generalized Stirling1 triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:          1;\n  1:          2,           4;\n  2:         10,          28,          12;\n  3:         80,         264,         180,         32;\n  4:        880,        3224,        2772,        832,         80;\n  5:      12320,       48656,       48480,      19040,       3200,       192;\n  6:     209440,      876432,      970128,     452960,     102000,     10944,      448;\n  7:    4188800,    18366400,    22031856,   11646208,    3172400,    463680,    34496,    1024;\n  8:   96342400,   439182400,   561831888,  326614400,  102080720,  18278400,  1875328,  102400,   2304;\n  9: 2504902400, 11804112000, 15925176288, 9990192768, 3470634720, 720232128, 91408128, 6948864, 290304, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(12*n-16)*x+4*x^2)*P(n-2,x) + (6*n-5+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (12*n - 16)*x + 4*x^2)*P[n - 2, x] + (6*n - 5 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (12*n - 16)*x + 4*x^2)*v[i - 2] + (6*n - 5 + 4*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-260e022530e5976c","steps":["basis A013609","\u03b8+1"],"edges":["e-dabb8392a3e47d84","e-a8130e28595215ba"],"text":"A286718 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A286718 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}