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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,          16;\n  2:         10,         112,          144;\n  3:         80,        1056,         2160,         1024;\n  4:        880,       12896,        33264,        26624,         6400;\n  5:      12320,      194624,       581760,       609280,       256000,        36864;\n  6:     209440,     3505728,     11641536,     14494720,      8160000,      2101248,      200704;\n  7:    4188800,    73465600,    264382272,    372678656,    253792000,     89026560,    15454208,    1048576;\n  8:   96342400,  1756729600,   6741982656,  10451660800,   8166457600,   3509452800,   840146944,  104857600,   5308416;\n  9: 2504902400, 47216448000, 191102115456, 319686168576, 277650777600, 138284568576, 40950841344, 7115636736, 668860416, 26214400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-24*n+16+(48*n-64)*x-144*x^2)*P(n-2,x) + ((48*n-64)*x^2+64*x^3)*d/dx P(n-2,x) + (6*n-5+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 16*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 24*n + 16 + (48*n - 64)*x - 144*x^2)*P[n - 2, x] + ((48*n - 64)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (6*n - 5 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], 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 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 24*n + 16 + (48*n - 64)*x - 144*x^2)*v[i - 2] + ((48*n - 64)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (6*n - 5 + 16*x)*v[i - 1] - 16*x^2*deriv(v[i - 1], x)); 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 A013611","(\u03b8+1)\u00b2"],"edges":["e-10584eb473514fc3","e-a0b0be8b64c50191"],"text":"A286718 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A286718 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}