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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9b8575e7966a7ec053d1bc5dcedaa52ffe3907975962314c63d6583a5abc2187","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cd3918419c5686a8","role":"sequence","name":"Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S1(j,k), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      1,       1;\n   3:      3,       5,        1;\n   4:     11,      27,       11,       1;\n   5:     49,     163,      102,      19,       1;\n   6:    261,    1109,      946,     270,      29,      1;\n   7:   1631,    8481,     9215,    3570,     585,     41,     1;\n   8:  11743,   72415,    95901,   47135,   10445,   1113,    55,    1;\n   9:  95901,  685007,  1072036,  640101,  179410,  25781,  1932,   71,  1;\n  10: 876809, 7124073, 12878132, 9063404, 3078726, 558334, 56308, 3132, 89, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-3)*P(n-2,x) + (n+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cd3918419c5686a8","steps":["basis A008277"],"edges":["e-b8d5969db6d485d6"],"text":"A269951 \u2192 basis A008277"},"headline":"A269951 \u2192 basis A008277","chain":null}