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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":"f7419ac899a1af78f6078dd76c3b86de505a1576fd3b8aa5d9f23bda0a7e0b70","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-4438653f9e73a43c","role":"sequence","name":"reversal of Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.","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,     2;\n  2:   6,    15,      5;\n  3:  11,    56,     68,      15;\n  4:  20,   171,    403,     317,      52;\n  5:  37,   484,   1817,    2707,    1559,      203;\n  6:  70,  1331,   7243,   16768,   18021,     8138,      877;\n  7: 135,  3632,  27162,   88827,  146129,   121886,    45093,    4140;\n  8: 264,  9939,  98831,  432015,  988326,  1249015,   847029,  264671,   21147;\n  9: 521, 27428, 354875, 2001115, 6019192, 10468201, 10676945, 6079691, 1640930, 115975;\n","recurrence_sentence":"The row polynomials satisfy 6*P(n,x) = (2*n-21+(21*n-97)*x-(6*n^2-72*n+167)*x^2-(12*n^2-72*n+90)*x^3-(6*n^2-17*n+2)*x^4-(4*n-8)*x^5)*P(n-2,x) - (3-(n-26)*x-(10*n-76)*x^2-(24*n-92)*x^3-(23*n-47)*x^4-(8*n-3)*x^5-4*x^6)*d/dx P(n-2,x) - (x+5*x^2+10*x^3+11*x^4+7*x^5+2*x^6)*d^2/dx^2 P(n-2,x) + (17-(12*n-55)*x-(12*n-19)*x^2-4*x^3)*P(n-1,x) + (1+9*x+15*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 21 + (21*n - 97)*x - (6*n^2 - 72*n + 167)*x^2 - (12*n^2 - 72*n + 90)*x^3 - (6*n^2 - 17*n + 2)*x^4 - (4*n - 8)*x^5)*P[n - 2, x] - (3 - (n - 26)*x - (10*n - 76)*x^2 - (24*n - 92)*x^3 - (23*n - 47)*x^4 - (8*n - 3)*x^5 - 4*x^6)*D[P[n - 2, x], x] - (x + 5*x^2 + 10*x^3 + 11*x^4 + 7*x^5 + 2*x^6)*D[P[n - 2, x], {x, 2}] + (17 - (12*n - 55)*x - (12*n - 19)*x^2 - 4*x^3)*P[n - 1, x] + (1 + 9*x + 15*x^2 + 8*x^3)*D[P[n - 1, x], x])/(6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 21 + (21*n - 97)*x - (6*n^2 - 72*n + 167)*x^2 - (12*n^2 - 72*n + 90)*x^3 - (6*n^2 - 17*n + 2)*x^4 - (4*n - 8)*x^5)*v[i - 2] - (3 - (n - 26)*x - (10*n - 76)*x^2 - (24*n - 92)*x^3 - (23*n - 47)*x^4 - (8*n - 3)*x^5 - 4*x^6)*deriv(v[i - 2], x) - (x + 5*x^2 + 10*x^3 + 11*x^4 + 7*x^5 + 2*x^6)*deriv(deriv(v[i - 2], x), x) + (17 - (12*n - 55)*x - (12*n - 19)*x^2 - 4*x^3)*v[i - 1] + (1 + 9*x + 15*x^2 + 8*x^3)*deriv(v[i - 1], x))/(6)); 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-f277d949034445ba","steps":["x \u21a6 x+1 (shift +1)","reverse","D+1"],"edges":["e-236d0fb15dc6eca0","e-0bbf9a609051b56f","e-fff386b09b3793e3"],"text":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse \u2192 D+1"},"headline":"A008278 \u2192 x \u21a6 x+1 (shift +1) \u2192 reverse \u2192 D+1","chain":null}