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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"3892b1e44e3262610b43fcdf278a5208c3e676f3e226504bdadda7f51e06cae3","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-df53153327eff8f9","role":"sequence","name":"Triangle T(n,k) read by rows, arising in enumeration of catafusenes.","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:     4,       5,        1;\n  3:    16,      47,       25,         3;\n  4:    64,     377,      391,       114,         9;\n  5:   256,    2831,     5041,      2655,       486,       27;\n  6:  1024,   20585,    58903,     49638,     15741,     1971,       81;\n  7:  4096,  147167,   650785,    822003,    400770,    84672,     7695,     243;\n  8: 16384, 1042457,  6949351,  12638394,   8878329,  2811078,   423306,   29160,    729;\n  9: 65536, 7346351, 72620881, 185147175, 179968446, 80045469, 17745966, 1998918, 107892, 2187;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (4*n-8+(3*n-6)*x)*P(n-1,x) + (3*n-6)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 5*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 4 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 8 + (3*n - 6)*x)*P[n - 1, x] + (3*n - 6)*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] = 4 + 5*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((4*n - 8 + (3*n - 6)*x)*v[i - 1] + (3*n - 6)*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-df53153327eff8f9","steps":["basis A008277"],"edges":["e-532f770f8bddb662"],"text":"A024462 \u2192 basis A008277"},"headline":"A024462 \u2192 basis A008277","chain":null}