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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":"1df21a47860419be554a54c65be5d78230d921822c39a39bad02a1d21b80b589","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-4172472abe953cea","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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:          11,           8,          1;\n  3:         104,          89,         18,          1;\n  4:        1397,        1320,        338,         32,         1;\n  5:       24334,       24649,       7320,        910,        50,       1;\n  6:      520959,      556352,     183499,      27160,      2005,      72,      1;\n  7:    13232820,    14741873,    5265946,     882959,     78960,    3871,     98,     1;\n  8:   388972905,   448758832,  170819396,   31505376,   3235134,  194544,   6804,   128,   1;\n  9: 12987518490, 15435574641, 6191398176, 1232236844, 140706132, 9813846, 425040, 11148, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-56*n+49+(8*n-14)*x+x^2)*P(n-2,x) + (8*n-10+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 56*n + 49 + (8*n - 14)*x + x^2)*P[n - 2, x] + (8*n - 10 + 2*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 56*n + 49 + (8*n - 14)*x + x^2)*v[i - 2] + (8*n - 10 + 2*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-4172472abe953cea","steps":["D+1"],"edges":["e-2bfd8476d75b734c"],"text":"A290319 \u2192 D+1"},"headline":"A290319 \u2192 D+1","chain":null}