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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":"f4c1abbdc0b18d819525173b3f8ec7733a7cf6666bfd0886b21fc23a3f47d64c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f425461cbf5769b3","role":"sequence","name":"Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        7,          4;\n   3:       49,        104,          32;\n   4:      343,       1660,        1824,          384;\n   5:     2401,      22160,       51392,        38400,          6144;\n   6:    16807,     271924,     1106880,      1678080,        952320,        122880;\n   7:   117649,    3190904,    20821472,     52999680,      59566080,      27279360,       2949120;\n   8:   823543,   36505900,   362441184,   1411127424,    2590586880,    2311249920,     887685120,      82575360;\n   9:  5764801,  411429920,  6014141312,  33917506560,   92531625984,  132347658240,   97851801600,   32369541120,    2642411520;\n  10: 40353607, 4594854244, 96748932480, 763092764160, 2926750740480, 6100311490560, 7133272473600, 4501678325760, 1307993702400, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(21+36*x+4*x^2-16*x^3)*P(n-2,x) - (12*x+12*x^2-16*x^3-16*x^4)*d/dx P(n-2,x) + (10+12*x-4*x^2)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 7 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(21 + 36*x + 4*x^2 - 16*x^3)*P[n - 2, x] - (12*x + 12*x^2 - 16*x^3 - 16*x^4)*D[P[n - 2, x], x] + (10 + 12*x - 4*x^2)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 7 + 4*x; for(n = 3, N, my(i = n); v[i] = -(21 + 36*x + 4*x^2 - 16*x^3)*v[i - 2] - (12*x + 12*x^2 - 16*x^3 - 16*x^4)*deriv(v[i - 2], x) + (10 + 12*x - 4*x^2)*v[i - 1] + (4*x + 4*x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f425461cbf5769b3","steps":["D+1"],"edges":["e-6aaaf313b1b0af89"],"text":"A225473 \u2192 D+1"},"headline":"A225473 \u2192 D+1","chain":null}