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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"152185340cdd70f1bd09558ed2eb517fb55866a04c07a40a82ff55147253ae67","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-b58d0e199d27164c","role":"sequence","name":"Pascal (1,6) 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:    6;\n  1:    7,     6;\n  2:   14,    25,      6;\n  3:   28,    89,     49,       6;\n  4:   56,   295,    285,      79,      6;\n  5:  112,   941,   1435,     680,    115,      6;\n  6:  224,  2935,   6681,    4835,   1370,    157,     6;\n  7:  448,  9029,  29659,   30856,  13055,   2469,   205,    6;\n  8:  896, 27535, 127665,  183939, 109186,  30338,  4109,  259,   6;\n  9: 1792, 83501, 538195, 1047360, 839055, 321552, 63210, 6440, 319, 6;\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) = 6, P(1,x) = 7 + 6*x.","mathematica":"P[0, x_] := 6;\nP[1, x_] := 7 + 6*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] = 6; v[2] = 7 + 6*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-b58d0e199d27164c","steps":["basis A008277"],"edges":["e-770b285f19415f37"],"text":"A096956 \u2192 basis A008277"},"headline":"A096956 \u2192 basis A008277","chain":null}