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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"a79ff980900f5f6147c2250694e0ec051011a1c2f648da98dbbc9f5b4a7630de","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c5cd4d1a5bdd7e74","role":"sequence","name":"Pascal (1,3) 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:    3;\n  1:    4,     3;\n  2:    8,    13,      3;\n  3:   16,    47,     25,      3;\n  4:   32,   157,    147,     40,      3;\n  5:   64,   503,    745,    347,     58,      3;\n  6:  128,  1573,   3483,   2480,    695,     79,     3;\n  7:  256,  4847,  15505,  15883,   6650,   1248,   103,    3;\n  8:  512, 14797,  66867,  94920,  55783,  15386,  2072,  130,   3;\n  9: 1024, 44903, 282265, 541467, 429618, 163485, 31962, 3242, 160, 3;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) + (n-1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 3;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] + (n - 1)*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] + (n - 1)*x*deriv(v[i - 1], x))/(n - 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-c5cd4d1a5bdd7e74","steps":["basis A008277"],"edges":["e-d614fa73ffeda478"],"text":"A095660 \u2192 basis A008277"},"headline":"A095660 \u2192 basis A008277","chain":null}