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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e73bb1a117104416f74c6d9996489e91a138059bcaacfe633879378fc22a97cf","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-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 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:          1,           3;\n  2:          2,           9,           9;\n  3:         10,          51,          72,          27;\n  4:        100,         540,         873,         486,          81;\n  5:       1700,        9480,       16461,       10881,        2835,        243;\n  6:      44200,      251580,      456426,      332289,      106353,      14823,       729;\n  7:    1635400,     9441060,    17642502,    13663971,     4931928,     867510,     71442,     2187;\n  8:   81770000,   476959200,   910448280,   736126056,   287588313,   58171284,   6174630,   323676,    6561;\n  9: 5315050000, 31247658000, 60610015800, 50579538480, 20901618513, 4643898399, 575864802, 39562830, 1397493, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-2*n+2+3*x)*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 - 2*n + 2 + 3*x)*P[n - 1, 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 - 2*n + 2 + 3*x)*v[i - 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-54284d24a094e2d3","steps":["basis A013610"],"edges":["e-6064ce9102e0184f"],"text":"A269944 \u2192 basis A013610"},"headline":"A269944 \u2192 basis A013610","chain":null}