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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":"d3381b4eb4bd0b86fa9ae7fd09e8adef330051af6636147de96a5af4ed7a0014","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-994dbf40e5f5e735","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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:        3,          2;\n  2:       13,         24,           6;\n  3:       70,        279,         177,           24;\n  4:      446,       3535,        4305,         1400,          120;\n  5:     3276,      49965,      104705,        61400,        12170,          720;\n  6:    27252,     789621,     2690765,      2536100,       872480,       116424,        5040;\n  7:   253296,   13891689,    74468807,    106068690,     56358365,     12724908,     1220436,      40320;\n  8:  2602224,  270294921,  2232714407,   4627581826,   3564813805,   1211143164,   193024244,   13940352,    362880;\n  9: 29288160, 5776027245, 72546228995, 213388039480, 229120729999, 109885000974, 25873597568, 3063592224, 172535184, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(3*n^3-4*n^2+n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) - ((3*n^3-4*n^2+n)*x+(2*n^3-4*n^2+2*n)*x^2)*d/dx P(n-2,x) - (n^3-2*n^2+n)*x^2*d^2/dx^2 P(n-2,x) + (2*n^2-1+(2*n^2-2*n)*x)*P(n-1,x) + (2*n^2-2*n)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (3*n^3 - 4*n^2 + n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] - ((3*n^3 - 4*n^2 + n)*x + (2*n^3 - 4*n^2 + 2*n)*x^2)*D[P[n - 2, x], x] - (n^3 - 2*n^2 + n)*x^2*D[P[n - 2, x], {x, 2}] + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*P[n - 1, x] + (2*n^2 - 2*n)*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (3*n^3 - 4*n^2 + n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] - ((3*n^3 - 4*n^2 + n)*x + (2*n^3 - 4*n^2 + 2*n)*x^2)*deriv(v[i - 2], x) - (n^3 - 2*n^2 + n)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 1 + (2*n^2 - 2*n)*x)*v[i - 1] + (2*n^2 - 2*n)*x*deriv(v[i - 1], x))/(n - 1)); 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-4d60c3dd8627c01d","steps":["\u03b8+1","basis A008277"],"edges":["e-8baeeca6d59e95f0","e-423487dba3f1c73a"],"text":"A094638 \u2192 \u03b8+1 \u2192 basis A008277"},"headline":"A094638 \u2192 \u03b8+1 \u2192 basis A008277","chain":null}