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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":"0dca3161226c9da17db61f254955b7e5cebd6288da6899ffffee386d47bc26bc","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-5c0c5bce3aee8449","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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:  1;\n   3:  4,    2;\n   4:  9,   20,     3;\n   5: 16,   84,    60,      4;\n   6: 25,  240,   420,    140,      5;\n   7: 36,  550,  1800,   1512,    280,      6;\n   8: 49, 1092,  5775,   9240,   4410,    504,     7;\n   9: 64, 1960, 15288,  40040,  36960,  11088,   840,    8;\n  10: 81, 3264, 35280, 137592, 210210, 123552, 24948, 1320, 9;\n","recurrence_sentence":"The row polynomials satisfy (8*n^5-40*n^4+58*n^3-8*n^2-24*n)*P(n,x) = (8*n^5-24*n^4+18*n^3+4*n^2-6*n+(72*n^5-548*n^4+1580*n^3-2154*n^2+1410*n-360)*x)*P(n-1,x) + ((16*n^4-96*n^3+190*n^2-158*n+48)*x-(104*n^4-608*n^3+1202*n^2-1010*n+312)*x^2)*d/dx P(n-1,x) - ((10*n^3-31*n^2+33*n-12)*x^2-(40*n^3-124*n^2+132*n-48)*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^5 - 24*n^4 + 18*n^3 + 4*n^2 - 6*n + (72*n^5 - 548*n^4 + 1580*n^3 - 2154*n^2 + 1410*n - 360)*x)*P[n - 1, x] + ((16*n^4 - 96*n^3 + 190*n^2 - 158*n + 48)*x - (104*n^4 - 608*n^3 + 1202*n^2 - 1010*n + 312)*x^2)*D[P[n - 1, x], x] - ((10*n^3 - 31*n^2 + 33*n - 12)*x^2 - (40*n^3 - 124*n^2 + 132*n - 48)*x^3)*D[P[n - 1, x], {x, 2}])/(8*n^5 - 40*n^4 + 58*n^3 - 8*n^2 - 24*n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((8*n^5 - 24*n^4 + 18*n^3 + 4*n^2 - 6*n + (72*n^5 - 548*n^4 + 1580*n^3 - 2154*n^2 + 1410*n - 360)*x)*v[i - 1] + ((16*n^4 - 96*n^3 + 190*n^2 - 158*n + 48)*x - (104*n^4 - 608*n^3 + 1202*n^2 - 1010*n + 312)*x^2)*deriv(v[i - 1], x) - ((10*n^3 - 31*n^2 + 33*n - 12)*x^2 - (40*n^3 - 124*n^2 + 132*n - 48)*x^3)*deriv(deriv(v[i - 1], x), x))/(8*n^5 - 40*n^4 + 58*n^3 - 8*n^2 - 24*n)); 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-79ebafb266da3b41","steps":["\u2299 C(n,k+1) (shift -1)","D"],"edges":["e-1491609a6fc5f8c9","e-2aa190e7e347a111"],"text":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 D"},"headline":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 D","chain":null}