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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"612c5c949f15f1c21ef602748566d2c6173c7bc19c6f453c7c4aed2a0475906c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3dd184aba99388f7","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","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:        10,         1;\n  2:        99,        20,         1;\n  3:       972,       297,        30,        1;\n  4:      9477,      3888,       594,       40,        1;\n  5:     91854,     47385,      9720,      990,       50,       1;\n  6:    885735,    551124,    142155,    19440,     1485,      60,     1;\n  7:   8503056,   6200145,   1928934,   331695,    34020,    2079,    70,    1;\n  8:  81310473,  68024448,  24800580,  5143824,   663390,   54432,  2772,   80,  1;\n  9: 774840978, 731794257, 306110016, 74401740, 11573604, 1194102, 81648, 3564, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+8+x)*P(n,x) = (9*n+81+(n+18)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n + 81 + (n + 18)*x + x^2)*P[n - 1, x])/(n + 8 + 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] = ((9*n + 81 + (n + 18)*x + x^2)*v[i - 1])/(n + 8 + 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-3dd184aba99388f7","steps":["D+1"],"edges":["e-a3ff94c4a2b65d05"],"text":"A038291 \u2192 D+1"},"headline":"A038291 \u2192 D+1","chain":null}