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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":"c013bc70be917d53ef822ca9e1d536d319dbc0b91cdd7c822a4438a2f7f9c0a9","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-aac530d688cc3150","role":"sequence","name":"Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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:      18,         19,           1;\n  3:     119,        354,         121,            1;\n  4:     730,       6091,        4868,          733,            1;\n  5:    4401,      87890,      166980,        59345,         4405,            1;\n  6:   26432,    1136911,     4602975,      3570205,       687395,        26437,           1;\n  7:  158623,   13792314,   107186901,    167283425,     67540915,      7768578,      158629,         1;\n  8:  951774,  161021259,  2225998211,   6363029136,   4988532775,   1195529614,    86697154,    951781,       1;\n  9: 5710685, 1836588906, 42796338388, 207624701263, 290496602256, 132177898930, 20368373536, 961124583, 5710693, 1;\n","recurrence_sentence":"The row polynomials satisfy 40*P(n,x) = (750*n^2-2450*n+1780+(375*n^2-1225*n-610)*x-1750*x^2-(1875*n^2-8000*n+7200)*x^3)*P(n-4,x) + ((375*n^2-1225*n+890)*x-750*x^2-(1875*n^2-8000*n+7200)*x^3)*d/dx P(n-4,x) + (750*n^3-5225*n^2+10845*n-6586+(375*n^3-1525*n^2-1130*n+8638)*x-(375*n^2+1150*n-6760)*x^2-(1875*n^3-13250*n^2+29600*n-18160)*x^3)*P(n-3,x) + (825*n^2-2695*n+1958+(375*n^3-3400*n^2+7995*n-6812)*x-(750*n^2-2075*n-2020)*x^2-(1875*n^3-15125*n^2+37600*n-28110)*x^3+(1875*n^2-8000*n+7200)*x^4)*d/dx P(n-3,x) + ((375*n^2-1225*n+890)*x-(375*n^2-1225*n+1640)*x^2-(1875*n^2-8000*n+6450)*x^3+(1875*n^2-8000*n+7200)*x^4)*d^2/dx^2 P(n-3,x) - (75*n^2+655*n-902+(1500*n^2-8750*n+8990)*x+(625*n^2-3500*n+4900)*x^2+(2000*n-3600)*x^3)*P(n-2,x) - (75*n^2-245*n+178+(750*n-3250)*x-(375*n^2-850*n-3160)*x^2+500*x^3-2000*x^4)*d/dx P(n-2,x) + (1000*x^2-2000*x^3+1000*x^4)*d^2/dx^2 P(n-2,x) + (150*n+60+(400*n-1420)*x+400*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 18 + 19*x + x^2, P(3,x) = 119 + 354*x + 121*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 18 + 19*x + x^2;\nP[3, x_] := 119 + 354*x + 121*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((750*n^2 - 2450*n + 1780 + (375*n^2 - 1225*n - 610)*x - 1750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*P[n - 4, x] + ((375*n^2 - 1225*n + 890)*x - 750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*D[P[n - 4, x], x] + (750*n^3 - 5225*n^2 + 10845*n - 6586 + (375*n^3 - 1525*n^2 - 1130*n + 8638)*x - (375*n^2 + 1150*n - 6760)*x^2 - (1875*n^3 - 13250*n^2 + 29600*n - 18160)*x^3)*P[n - 3, x] + (825*n^2 - 2695*n + 1958 + (375*n^3 - 3400*n^2 + 7995*n - 6812)*x - (750*n^2 - 2075*n - 2020)*x^2 - (1875*n^3 - 15125*n^2 + 37600*n - 28110)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*D[P[n - 3, x], x] + ((375*n^2 - 1225*n + 890)*x - (375*n^2 - 1225*n + 1640)*x^2 - (1875*n^2 - 8000*n + 6450)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*D[P[n - 3, x], {x, 2}] - (75*n^2 + 655*n - 902 + (1500*n^2 - 8750*n + 8990)*x + (625*n^2 - 3500*n + 4900)*x^2 + (2000*n - 3600)*x^3)*P[n - 2, x] - (75*n^2 - 245*n + 178 + (750*n - 3250)*x - (375*n^2 - 850*n - 3160)*x^2 + 500*x^3 - 2000*x^4)*D[P[n - 2, x], x] + (1000*x^2 - 2000*x^3 + 1000*x^4)*D[P[n - 2, x], {x, 2}] + (150*n + 60 + (400*n - 1420)*x + 400*x^2)*P[n - 1, x])/(40)]];\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; v[3] = 18 + 19*x + x^2; v[4] = 119 + 354*x + 121*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((750*n^2 - 2450*n + 1780 + (375*n^2 - 1225*n - 610)*x - 1750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*v[i - 4] + ((375*n^2 - 1225*n + 890)*x - 750*x^2 - (1875*n^2 - 8000*n + 7200)*x^3)*deriv(v[i - 4], x) + (750*n^3 - 5225*n^2 + 10845*n - 6586 + (375*n^3 - 1525*n^2 - 1130*n + 8638)*x - (375*n^2 + 1150*n - 6760)*x^2 - (1875*n^3 - 13250*n^2 + 29600*n - 18160)*x^3)*v[i - 3] + (825*n^2 - 2695*n + 1958 + (375*n^3 - 3400*n^2 + 7995*n - 6812)*x - (750*n^2 - 2075*n - 2020)*x^2 - (1875*n^3 - 15125*n^2 + 37600*n - 28110)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*deriv(v[i - 3], x) + ((375*n^2 - 1225*n + 890)*x - (375*n^2 - 1225*n + 1640)*x^2 - (1875*n^2 - 8000*n + 6450)*x^3 + (1875*n^2 - 8000*n + 7200)*x^4)*deriv(deriv(v[i - 3], x), x) - (75*n^2 + 655*n - 902 + (1500*n^2 - 8750*n + 8990)*x + (625*n^2 - 3500*n + 4900)*x^2 + (2000*n - 3600)*x^3)*v[i - 2] - (75*n^2 - 245*n + 178 + (750*n - 3250)*x - (375*n^2 - 850*n - 3160)*x^2 + 500*x^3 - 2000*x^4)*deriv(v[i - 2], x) + (1000*x^2 - 2000*x^3 + 1000*x^4)*deriv(deriv(v[i - 2], x), x) + (150*n + 60 + (400*n - 1420)*x + 400*x^2)*v[i - 1])/(40)); 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-aac530d688cc3150","steps":["D+1"],"edges":["e-8cf67d69c418ddd8"],"text":"A144442 \u2192 D+1"},"headline":"A144442 \u2192 D+1","chain":null}