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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:          4,           2;\n  2:         24,          22,           4;\n  3:        212,         252,          84,          8;\n  4:       2492,        3460,        1548,        272,         16;\n  5:      36648,       56648,       30440,       7320,        800,        32;\n  6:     647656,     1084968,      663096,     194840,      29520,      2208,       64;\n  7:   13372000,    23871104,    16078608,    5449472,    1020880,    106848,     5824,     128;\n  8:  315933600,   594145792,   432238096,  163318176,   35648144,   4653824,   357504,   14848,   256;\n  9: 8406958400, 16518840192, 12801036672, 5274055968, 1294320384, 198388512, 19138560, 1126656, 36864, 512;\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) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*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] = 4 + 2*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","D+1"],"edges":["e-dabb8392a3e47d84","e-437301389b801153"],"text":"A286718 \u2192 basis A013609 \u2192 D+1"},"headline":"A286718 \u2192 basis A013609 \u2192 D+1","chain":null}