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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":"263afd591ee5b482444d2fe5766f2378fc4fb8919d8d43c000ce136225488843","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-f268708db386b729","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= 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:    1;\n   2:    3,     1;\n   3:    9,     7,     1;\n   4:   24,    34,    12,     1;\n   5:   60,   126,    83,    18,     1;\n   6:  144,   404,   401,   165,    25,     1;\n   7:  336,  1184,  1587,   997,   290,    33,    1;\n   8:  768,  3264,  5536,  4740,  2128,   469,   42,    1;\n   9: 1728,  8608, 17704, 19308, 11909,  4088,  714,   52,  1;\n  10: 3840, 21952, 53136, 70616, 55807, 26505, 7266, 1038, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-12*n^2+15*n-6+(4*n^3-10*n^2+6)*x+(8*n^2-22*n+14)*x^2+(4*n-4)*x^3)*P(n,x) = (6*n^3-18*n^2+12*n+(17*n^3-46*n^2+13*n+16)*x+(12*n^3-14*n^2-42*n+48)*x^2+(24*n^2-62*n+42)*x^3+(12*n-20)*x^4)*P(n-1,x) - ((6*n^2-6*n)*x+(14*n^2-6*n-8)*x^2+(8*n^2+16*n-20)*x^3+(16*n-4)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^3 - 18*n^2 + 12*n + (17*n^3 - 46*n^2 + 13*n + 16)*x + (12*n^3 - 14*n^2 - 42*n + 48)*x^2 + (24*n^2 - 62*n + 42)*x^3 + (12*n - 20)*x^4)*P[n - 1, x] - ((6*n^2 - 6*n)*x + (14*n^2 - 6*n - 8)*x^2 + (8*n^2 + 16*n - 20)*x^3 + (16*n - 4)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 15*n - 6 + (4*n^3 - 10*n^2 + 6)*x + (8*n^2 - 22*n + 14)*x^2 + (4*n - 4)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((6*n^3 - 18*n^2 + 12*n + (17*n^3 - 46*n^2 + 13*n + 16)*x + (12*n^3 - 14*n^2 - 42*n + 48)*x^2 + (24*n^2 - 62*n + 42)*x^3 + (12*n - 20)*x^4)*v[i - 1] - ((6*n^2 - 6*n)*x + (14*n^2 - 6*n - 8)*x^2 + (8*n^2 + 16*n - 20)*x^3 + (16*n - 4)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 15*n - 6 + (4*n^3 - 10*n^2 + 6)*x + (8*n^2 - 22*n + 14)*x^2 + (4*n - 4)*x^3)); 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-2821c21537bbbcdd","steps":["reverse","D+1"],"edges":["e-23a5df9ffca078bb","e-32bd4aa1cdd8d397"],"text":"A056242 \u2192 reverse \u2192 D+1"},"headline":"A056242 \u2192 reverse \u2192 D+1","chain":null}