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h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"d70e87622d0597a875f491e2f94f9c61eb040e9149312063a09dd717dccc8fa0","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-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:           10,              17,                4;\n  3:          100,             413,              265,               36;\n  4:         1700,           15229,            19593,             6580,              576;\n  5:        44200,          819179,          1869793,          1154405,           237076,            14400;\n  6:      1635400,        61391267,        233297881,        234701273,         84469336,         11659536,          518400;\n  7:     81770000,      6157870033,      37536258471,      57667746950,      32279819033,       7578560584,       749646864,        25401600;\n  8:   5315050000,    799598514257,    7638601567015,   17222931508294,   14052557714393,    4983654242952,    821619319312,     61008420096,    1625702400;\n  9: 435834100000, 130765076873891, 1928586262006477, 6216179467123765, 7100395884215372, 3565296791183457, 870355767048328, 106145629506576, 6128444779776, 131681894400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2+1+n^2*x)*P(n-1,x) + n^2*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n^2 + 1 + n^2*x)*P[n - 1, x] + n^2*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; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + 1 + n^2*x)*v[i - 1] + n^2*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-e734a631f8279507","steps":["basis A008277"],"edges":["e-e6241a3fc3f0241e"],"text":"A008955 \u2192 basis A008277"},"headline":"A008955 \u2192 basis A008277","chain":null}