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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"f7ad78d93cae6189b815afd8f2fd57bfd220b011597167e192fcef2625c2e40f","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-21854f4257effb01","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(k,n-k).","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:  3,    5,     1;\n  3:  5,   17,     9,      1;\n  4:  8,   52,    52,     14,      1;\n  5: 13,  151,   252,    121,     20,     1;\n  6: 21,  427,  1115,    844,    240,    27,     1;\n  7: 34, 1190,  4679,   5227,   2265,   428,    35,    1;\n  8: 55, 3289, 18999,  30078,  18596,  5235,   707,   44,  1;\n  9: 89, 9047, 75513, 164898, 139610, 54839, 10857, 1102, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + x*d/dx P(n-2,x) + (1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (1 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\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; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (1 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-21854f4257effb01","steps":["basis A008277"],"edges":["e-e511d50f371cf4d8"],"text":"A030528 \u2192 basis A008277"},"headline":"A030528 \u2192 basis A008277","chain":null}