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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"ad88a6d6b5050878285f8ec14d10ab11a47f8b8d1d2c807cfbc111ed05c555c6","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-01d7f2ed2a619101","role":"sequence","name":"basis_map_A013611 of Square the entries of Pascal's 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:         6,           4;\n  2:        46,          72,          16;\n  3:       396,        1144,         576,           64;\n  4:      3606,       17680,       15520,         3840,         256;\n  5:     33876,      270424,      373760,       156800,       23040,        1024;\n  6:    324556,     4119024,     8487136,      5411840,     1326080,      129024,       4096;\n  7:   3151896,    62678704,   186123392,    169961344,    62003200,     9949184,     688128,     16384;\n  8:  30915046,   954568224,  3993574720,   5030462976,  2567502336,   605036544,   68468736,   3538944,    65536;\n  9: 305543556, 14564110744, 84494384640, 143112720640, 98254205952, 31901657088, 5250416640, 441384960, 17694720, 262144;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(16*n-16+(48*n-48)*x+(16*n-16)*x^2)*P(n-2,x) - ((48*n-48)*x+(32*n-32)*x^2)*d/dx P(n-2,x) - (16*n-16)*x^2*d^2/dx^2 P(n-2,x) + (12*n-6+(8*n-4)*x)*P(n-1,x) + (8*n-4)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(16*n - 16 + (48*n - 48)*x + (16*n - 16)*x^2)*P[n - 2, x] - ((48*n - 48)*x + (32*n - 32)*x^2)*D[P[n - 2, x], x] - (16*n - 16)*x^2*D[P[n - 2, x], {x, 2}] + (12*n - 6 + (8*n - 4)*x)*P[n - 1, x] + (8*n - 4)*x*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(16*n - 16 + (48*n - 48)*x + (16*n - 16)*x^2)*v[i - 2] - ((48*n - 48)*x + (32*n - 32)*x^2)*deriv(v[i - 2], x) - (16*n - 16)*x^2*deriv(deriv(v[i - 2], x), x) + (12*n - 6 + (8*n - 4)*x)*v[i - 1] + (8*n - 4)*x*deriv(v[i - 1], x))/(n)); 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-b13531ad79e7c629","steps":["basis A013611","basis A008277"],"edges":["e-52e1bd34b019f2ca","e-682df92e56c02b05"],"text":"A008459 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A008459 \u2192 basis A013611 \u2192 basis A008277","chain":null}