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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"f84a4f2cb0699929e90816b1f8847271d2c7720eedfc181b13d3e28ca7572eb3","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-10c99b3e7ebd1948","role":"sequence","name":"Elements in the 5-Pascal triangle (by row).","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,     1;\n  2:   7,     8,      1;\n  3:  14,    31,     12,      1;\n  4:  28,   107,     79,     17,      1;\n  5:  56,   349,    423,    164,     23,     1;\n  6: 112,  1103,   2041,   1243,    302,    30,     1;\n  7: 224,  3421,   9267,   8256,   3055,   512,    38,    1;\n  8: 448, 10487,  40489,  50547,  26586,  6639,   816,   47,  1;\n  9: 896, 31909, 172443, 293224, 210063, 73059, 13167, 1239, 57, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 7 + 8*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 7 + 8*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 7 + 8*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 2)); 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-10c99b3e7ebd1948","steps":["basis A008277"],"edges":["e-74168417bc03f911"],"text":"A028313 \u2192 basis A008277"},"headline":"A028313 \u2192 basis A008277","chain":null}