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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":"42c0bc7239eb7c50ebe73633e06d3e74f976feaf02f568e0daf02771663507fb","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-95c4930a6d47b128","role":"sequence","name":"basis_map_A013609 of Coefficients of a special case of Poisson-Charlier polynomials.","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:        0;\n   2:        2;\n   3:       10,         8;\n   4:       50,        88,        24;\n   5:      288,       856,       672,         64;\n   6:     1922,      8432,     12264,       6016,        160;\n   7:    14626,     86752,    191304,     185856,      67360,        384;\n   8:   125050,    945048,   2836704,    4224576,    3119360,     913536,        896;\n   9:  1187472,  10971120,  42000192,   85236416,   96661760,   58041600,   14407680,      2048;\n  10: 12408738, 136019808, 636161424, 1645056384, 2539208160, 2338438656, 1189090560, 257396736, 4608;\n","recurrence_sentence":"The row polynomials satisfy (4*n-8)*P(n,x) = -(4*n^3-36*n^2+116*n-132+(24*n^3-240*n^2+792*n-864)*x+(48*n^3-528*n^2+1776*n-1872)*x^2+(32*n^3-384*n^2+1312*n-1344)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(8*n^3-68*n^2+200*n-204)*x+(24*n^3-216*n^2+648*n-648)*x^2+(32*n^3-304*n^2+928*n-912)*x^3+(16*n^3-160*n^2+496*n-480)*x^4)*d/dx P(n-3,x) - (12*n^2-52*n+60+(48*n^2-208*n+224)*x+(48*n^2-192*n+192)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(6*n^3-42*n^2+96*n-68)*x+(12*n^3-96*n^2+236*n-184)*x^2+(8*n^3-72*n^2+192*n-160)*x^3)*d/dx P(n-2,x) + (8*n^2-24*n+20+(16*n^2-64*n+64)*x)*P(n-1,x) - (n^2-5*n+8+(4*n^2-24*n+36)*x+(4*n^2-28*n+40)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 2, P(3,x) = 10 + 8*x.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[3, x_] := 10 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 36*n^2 + 116*n - 132 + (24*n^3 - 240*n^2 + 792*n - 864)*x + (48*n^3 - 528*n^2 + 1776*n - 1872)*x^2 + (32*n^3 - 384*n^2 + 1312*n - 1344)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*D[P[n - 3, x], x] - (12*n^2 - 52*n + 60 + (48*n^2 - 208*n + 224)*x + (48*n^2 - 192*n + 192)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 24*n + 20 + (16*n^2 - 64*n + 64)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*D[P[n - 1, x], x])/(4*n - 8)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; v[3] = 10 + 8*x; for(n = 4, N, my(i = n); v[i] = (-(4*n^3 - 36*n^2 + 116*n - 132 + (24*n^3 - 240*n^2 + 792*n - 864)*x + (48*n^3 - 528*n^2 + 1776*n - 1872)*x^2 + (32*n^3 - 384*n^2 + 1312*n - 1344)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (8*n^3 - 68*n^2 + 200*n - 204)*x + (24*n^3 - 216*n^2 + 648*n - 648)*x^2 + (32*n^3 - 304*n^2 + 928*n - 912)*x^3 + (16*n^3 - 160*n^2 + 496*n - 480)*x^4)*deriv(v[i - 3], x) - (12*n^2 - 52*n + 60 + (48*n^2 - 208*n + 224)*x + (48*n^2 - 192*n + 192)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (6*n^3 - 42*n^2 + 96*n - 68)*x + (12*n^3 - 96*n^2 + 236*n - 184)*x^2 + (8*n^3 - 72*n^2 + 192*n - 160)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 24*n + 20 + (16*n^2 - 64*n + 64)*x)*v[i - 1] - (n^2 - 5*n + 8 + (4*n^2 - 24*n + 36)*x + (4*n^2 - 28*n + 40)*x^2)*deriv(v[i - 1], x))/(4*n - 8)); 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-c9dce8c25621fdea","steps":["basis A013609","D"],"edges":["e-0705555781f6cbb3","e-388b604976a70313"],"text":"A046716 \u2192 basis A013609 \u2192 D"},"headline":"A046716 \u2192 basis A013609 \u2192 D","chain":null}