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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":"aaf668d6a43c6715c97469342fd8785aef96f81dc3860aeb5b2e6f549801d9fb","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-4f8c7533ab8b37bb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,5,1) array.","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:      3,       1;\n  2:     18,      11,       1;\n  3:     70,      74,      19,       1;\n  4:    303,     450,     193,      27,      1;\n  5:   1197,    2335,    1505,     375,     35,      1;\n  6:   4756,   11497,    9920,    3725,    620,     43,     1;\n  7:  18396,   53252,   58202,   30030,   7600,    928,    51,    1;\n  8:  70605,  238308,  316038,  209322,  73410,  13620,  1299,   59,  1;\n  9: 267751, 1033485, 1615926, 1315434, 599262, 154578, 22275, 1733, 67, 1;\n","recurrence_sentence":"The row polynomials satisfy (12*n^2+12*n+(6*n^3+18*n^2+12*n)*x+(23*n^2+11*n)*x^2+(n^2+22*n)*x^3+n*x^4)*P(n,x) = -(72*n^3-12*n^2-252*n+120+(66*n^3+234*n^2-408*n-24)*x+(247*n^2+57*n-442)*x^2+(11*n^2+219*n-333)*x^3+(10*n-36)*x^4-x^5)*P(n-1,x) + (144*n^2+432*n+288+(288*n^2+1416*n+1416)*x+(150*n^2+1578*n+2556)*x^2+(6*n^2+641*n+2063)*x^3+(48*n+683)*x^4+(n+49)*x^5+x^6)*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[(-(72*n^3 - 12*n^2 - 252*n + 120 + (66*n^3 + 234*n^2 - 408*n - 24)*x + (247*n^2 + 57*n - 442)*x^2 + (11*n^2 + 219*n - 333)*x^3 + (10*n - 36)*x^4 - x^5)*P[n - 1, x] + (144*n^2 + 432*n + 288 + (288*n^2 + 1416*n + 1416)*x + (150*n^2 + 1578*n + 2556)*x^2 + (6*n^2 + 641*n + 2063)*x^3 + (48*n + 683)*x^4 + (n + 49)*x^5 + x^6)*D[P[n - 1, x], x])/(12*n^2 + 12*n + (6*n^3 + 18*n^2 + 12*n)*x + (23*n^2 + 11*n)*x^2 + (n^2 + 22*n)*x^3 + n*x^4)]];\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] = (-(72*n^3 - 12*n^2 - 252*n + 120 + (66*n^3 + 234*n^2 - 408*n - 24)*x + (247*n^2 + 57*n - 442)*x^2 + (11*n^2 + 219*n - 333)*x^3 + (10*n - 36)*x^4 - x^5)*v[i - 1] + (144*n^2 + 432*n + 288 + (288*n^2 + 1416*n + 1416)*x + (150*n^2 + 1578*n + 2556)*x^2 + (6*n^2 + 641*n + 2063)*x^3 + (48*n + 683)*x^4 + (n + 49)*x^5 + x^6)*deriv(v[i - 1], x))/(12*n^2 + 12*n + (6*n^3 + 18*n^2 + 12*n)*x + (23*n^2 + 11*n)*x^2 + (n^2 + 22*n)*x^3 + n*x^4)); 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-6ad01dc6e705cdb6","steps":["basis A007318","D+1"],"edges":["e-1ae1e4b2be227437","e-ee00d51ddb535384"],"text":"A081580 \u2192 basis A007318 \u2192 D+1"},"headline":"A081580 \u2192 basis A007318 \u2192 D+1","chain":null}