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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":"48f458f27fddcef13d14ed7f3de05b98630ce700f3a55911c7995a3fa7378b18","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1397bdceeefc7b3d","role":"sequence","name":"Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*n).","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:        5,         2;\n  2:       35,        32,         6;\n  3:      252,       378,       165,        20;\n  4:     1815,      3840,      2772,       784,        70;\n  5:    13013,     35750,     36630,     17160,      3570,       252;\n  6:    92820,    314496,    420420,    280280,     96525,     15840,      924;\n  7:   658920,   2658936,   4399668,   3836560,   1881880,    510510,    69069,     3432;\n  8:  4657983,  21829632,  43139880,  46667040,  29980860,  11583936,  2586584,   297440,   12870;\n  9: 32807775, 175215150, 403038108, 521694096, 416340540, 210815640, 67016040, 12697776, 1268982, 48620;\n","recurrence_sentence":"The row polynomials satisfy (8*n^4-8*n^3-2*n^2+2*n+(12*n^3-18*n^2+6*n)*x+(4*n^2-2*n)*x^2)*P(n,x) = (54*n^4-27*n^3-96*n^2+87*n-18+(48*n^4+43*n^3-218*n^2+157*n-30)*x+(2*n^4+69*n^3-99*n^2+44*n-6)*x^2+(3*n^3+20*n^2-25*n+6)*x^3+(n^2-n)*x^4)*P(n-1,x) - ((18*n^3+9*n^2-23*n+6)*x+(20*n^3+37*n^2-42*n+9)*x^2+(2*n^3+31*n^2-10*n)*x^3+(3*n^2+10*n-3)*x^4+n*x^5)*d/dx 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[((54*n^4 - 27*n^3 - 96*n^2 + 87*n - 18 + (48*n^4 + 43*n^3 - 218*n^2 + 157*n - 30)*x + (2*n^4 + 69*n^3 - 99*n^2 + 44*n - 6)*x^2 + (3*n^3 + 20*n^2 - 25*n + 6)*x^3 + (n^2 - n)*x^4)*P[n - 1, x] - ((18*n^3 + 9*n^2 - 23*n + 6)*x + (20*n^3 + 37*n^2 - 42*n + 9)*x^2 + (2*n^3 + 31*n^2 - 10*n)*x^3 + (3*n^2 + 10*n - 3)*x^4 + n*x^5)*D[P[n - 1, x], x])/(8*n^4 - 8*n^3 - 2*n^2 + 2*n + (12*n^3 - 18*n^2 + 6*n)*x + (4*n^2 - 2*n)*x^2)]];\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] = ((54*n^4 - 27*n^3 - 96*n^2 + 87*n - 18 + (48*n^4 + 43*n^3 - 218*n^2 + 157*n - 30)*x + (2*n^4 + 69*n^3 - 99*n^2 + 44*n - 6)*x^2 + (3*n^3 + 20*n^2 - 25*n + 6)*x^3 + (n^2 - n)*x^4)*v[i - 1] - ((18*n^3 + 9*n^2 - 23*n + 6)*x + (20*n^3 + 37*n^2 - 42*n + 9)*x^2 + (2*n^3 + 31*n^2 - 10*n)*x^3 + (3*n^2 + 10*n - 3)*x^4 + n*x^5)*deriv(v[i - 1], x))/(8*n^4 - 8*n^3 - 2*n^2 + 2*n + (12*n^3 - 18*n^2 + 6*n)*x + (4*n^2 - 2*n)*x^2)); 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-1397bdceeefc7b3d","steps":["D+1"],"edges":["e-6f795952623be221"],"text":"A357613 \u2192 D+1"},"headline":"A357613 \u2192 D+1","chain":null}