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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7959ebab0f188c8144e32fb8e2a0871105ad4cdb8de03e777c88805b335c9400","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-32779d2065847504","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.","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:     7,       4;\n  2:    32,      46,       10;\n  3:   120,     330,      170,       20;\n  4:   400,    1895,     1745,      470,       35;\n  5:  1232,    9541,    14035,     6510,     1085,       56;\n  6:  3584,   44044,    97524,    69440,    19460,     2212,      84;\n  7:  9984,  191220,   614796,   631680,   265020,    49812,    4116,    120;\n  8: 26880,  793245,  3616995,  5161590,  3044265,   841890,  113610,   7140,   165;\n  9: 70400, 3177295, 20216625, 39055390, 31135335, 11889570, 2330790, 236940, 11715, 220;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(2*n+2+(n+1)*x)*P(n-2,x) - (n+1)*x*d/dx P(n-2,x) + (3*n+4+(n+3)*x)*P(n-1,x) + (n+3)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 2 + (n + 1)*x)*P[n - 2, x] - (n + 1)*x*D[P[n - 2, x], x] + (3*n + 4 + (n + 3)*x)*P[n - 1, x] + (n + 3)*x*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 2 + (n + 1)*x)*v[i - 2] - (n + 1)*x*deriv(v[i - 2], x) + (3*n + 4 + (n + 3)*x)*v[i - 1] + (n + 3)*x*deriv(v[i - 1], x))/(n)); 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-32779d2065847504","steps":["basis A008277"],"edges":["e-e51fcf90a18be50b"],"text":"A347056 \u2192 basis A008277"},"headline":"A347056 \u2192 basis A008277","chain":null}