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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"1e844e6e48340e0cde6eb56b4a194f939dd996bf83d4a1137adb8f4690808681","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-fc7cbd97dfcf0d00","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n, k) if n < 3, otherwise T(n, k) = binomial(n, k) + 2*binomial(n-2, k-1).","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:  12,    27,     11,      1;\n  4:  24,    93,     71,     16,      1;\n  5:  48,   303,    377,    151,     22,     1;\n  6:  96,   957,   1811,   1132,    283,    29,     1;\n  7: 192,  2967,   8201,   7471,   2830,   486,    37,    1;\n  8: 384,  9093,  35771,  45556,  24451,  6232,   782,   46,  1;\n  9: 768, 27663, 152177, 263551, 192262, 68075, 12488, 1196, 56, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (2*n-6+(n-3)*x)*P(n-1,x) + (n-3)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 5*x + x^2, P(3,x) = 12 + 27*x + 11*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 4 + 5*x + x^2;\nP[3, x_] := 12 + 27*x + 11*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 6 + (n - 3)*x)*P[n - 1, x] + (n - 3)*x*D[P[n - 1, x], x])/(n - 3)]];\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; v[4] = 12 + 27*x + 11*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2*n - 6 + (n - 3)*x)*v[i - 1] + (n - 3)*x*deriv(v[i - 1], x))/(n - 3)); 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-fc7cbd97dfcf0d00","steps":["basis A008277"],"edges":["e-0104d750fd8745b9"],"text":"A147644 \u2192 basis A008277"},"headline":"A147644 \u2192 basis A008277","chain":null}