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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":"c11e89bf3e031660d1819f920bd46a24c307146f7ad78d91cee0223b0ed7cd6b","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-cbba416b5dae7a51","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= 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:       6,        3;\n  2:      32,       39,        9;\n  3:     171,      342,      189,        27;\n  4:     923,     2598,     2394,       810,        81;\n  5:    5028,    18345,    24120,     13770,      3240,      243;\n  6:   27598,   124029,   213570,    175905,     70470,    12393,      729;\n  7:  152421,   815052,  1742202,   1895670,   1105650,   333396,    45927,    2187;\n  8:  846101,  5250828, 13435128,  18269496,  14181804,  6273288,  1490076,  166212,   6561;\n  9: 4716738, 33339087, 99470916, 162774360, 158846184, 94119732, 33006204, 6377292, 590490, 19683;\n","recurrence_sentence":"The row polynomials satisfy (27*n^3+63*n^2+6*n-24+(54*n^2+81*n-54)*x+(27*n+54)*x^2)*P(n,x) = (189*n^3+279*n^2-12*n-24+(243*n^3+621*n^2-108*n-54)*x+(486*n^2+189*n+54)*x^2+243*n*x^3)*P(n-1,x) - (36*n^2+66*n+20+(162*n^2+369*n+126)*x+(162*n^2+621*n+288)*x^2+(324*n+324)*x^3+162*x^4)*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[((189*n^3 + 279*n^2 - 12*n - 24 + (243*n^3 + 621*n^2 - 108*n - 54)*x + (486*n^2 + 189*n + 54)*x^2 + 243*n*x^3)*P[n - 1, x] - (36*n^2 + 66*n + 20 + (162*n^2 + 369*n + 126)*x + (162*n^2 + 621*n + 288)*x^2 + (324*n + 324)*x^3 + 162*x^4)*D[P[n - 1, x], x])/(27*n^3 + 63*n^2 + 6*n - 24 + (54*n^2 + 81*n - 54)*x + (27*n + 54)*x^2)]];\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] = ((189*n^3 + 279*n^2 - 12*n - 24 + (243*n^3 + 621*n^2 - 108*n - 54)*x + (486*n^2 + 189*n + 54)*x^2 + 243*n*x^3)*v[i - 1] - (36*n^2 + 66*n + 20 + (162*n^2 + 369*n + 126)*x + (162*n^2 + 621*n + 288)*x^2 + (324*n + 324)*x^3 + 162*x^4)*deriv(v[i - 1], x))/(27*n^3 + 63*n^2 + 6*n - 24 + (54*n^2 + 81*n - 54)*x + (27*n + 54)*x^2)); 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-57b4b14604f4c0d7","steps":["basis A013610","D+1"],"edges":["e-125d456539fb62ba","e-c5f7e9862eca5d77"],"text":"A126216 \u2192 basis A013610 \u2192 D+1"},"headline":"A126216 \u2192 basis A013610 \u2192 D+1","chain":null}