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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"16e1e288aaaf2da4d2dc124af95ec66876d7df74bda33f4dacd7075e93e177a7","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-f3a4ae052af3da90","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 5.","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,           5;\n  2:       105,         130,           25;\n  3:      1040,        2745,         1290,          125;\n  4:      9695,       49330,        47340,        11570,           625;\n  5:     86310,      780005,      1393640,       681150,         98810,         3125;\n  6:    743605,    11221290,     34513095,     30742100,       8886555,       821730,        15625;\n  7:   6261340,   150728825,    749582370,   1128849275,     587133880,    109372095,      6730090,       78125;\n  8:  51887595,  1925926570,  14755917010,  35310934350,   30707345420,  10229522870,   1297693590,    54621970,    390625;\n  9: 425257010, 23719871925, 269777936160, 976521847740, 1335717945540, 738878961870, 167581205520, 15034715100, 440882010, 1953125;\n","recurrence_sentence":"The row polynomials satisfy (3*n-4+6*x)*P(n,x) = -(120*n-40+(90*n^2-195*n+295)*x+(27*n^3-81*n^2+270*n-352)*x^2+(81*n^2-162*n+135)*x^3+(54*n-18)*x^4)*P(n-2,x) - ((45*n-15)*x+(27*n^2-90*n+117)*x^2-(27*n^2-126*n+183)*x^3-(81*n-135)*x^4-54*x^5)*d/dx P(n-2,x) + (39*n-37+(18*n^2-39*n+98)*x+(45*n-51)*x^2+18*x^3)*P(n-1,x) + ((9*n-12)*x-(9*n-30)*x^2-18*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 5*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 10 + 5*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(120*n - 40 + (90*n^2 - 195*n + 295)*x + (27*n^3 - 81*n^2 + 270*n - 352)*x^2 + (81*n^2 - 162*n + 135)*x^3 + (54*n - 18)*x^4)*P[n - 2, x] - ((45*n - 15)*x + (27*n^2 - 90*n + 117)*x^2 - (27*n^2 - 126*n + 183)*x^3 - (81*n - 135)*x^4 - 54*x^5)*D[P[n - 2, x], x] + (39*n - 37 + (18*n^2 - 39*n + 98)*x + (45*n - 51)*x^2 + 18*x^3)*P[n - 1, x] + ((9*n - 12)*x - (9*n - 30)*x^2 - 18*x^3)*D[P[n - 1, x], x])/(3*n - 4 + 6*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 10 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(120*n - 40 + (90*n^2 - 195*n + 295)*x + (27*n^3 - 81*n^2 + 270*n - 352)*x^2 + (81*n^2 - 162*n + 135)*x^3 + (54*n - 18)*x^4)*v[i - 2] - ((45*n - 15)*x + (27*n^2 - 90*n + 117)*x^2 - (27*n^2 - 126*n + 183)*x^3 - (81*n - 135)*x^4 - 54*x^5)*deriv(v[i - 2], x) + (39*n - 37 + (18*n^2 - 39*n + 98)*x + (45*n - 51)*x^2 + 18*x^3)*v[i - 1] + ((9*n - 12)*x - (9*n - 30)*x^2 - 18*x^3)*deriv(v[i - 1], x))/(3*n - 4 + 6*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-f3a4ae052af3da90","steps":["D+1"],"edges":["e-b8039b9b0b256241"],"text":"A257624 \u2192 D+1"},"headline":"A257624 \u2192 D+1","chain":null}