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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"e1123610e550d29c306f93b6528573f93c71fa798c2d28e3bbfa8d42f7402e24","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-70594f2875298da0","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:         9,          6;\n  2:        63,        159,          54;\n  3:       459,       2163,        2826,          648;\n  4:      3654,      25848,       62631,        55998,          9720;\n  5:     32058,     306000,     1106667,      1827468,       1250964,        174960;\n  6:    309384,    3741516,    18182187,     44638803,      56283984,      31344084,       3674160;\n  7:   3268296,   48040236,   294863571,    969770043,    1812807864,    1856091348,     873505296,      88179840;\n  8:  37581840,  651991464,  4852592712,  20110498776,   50244947109,   76109272812,   65768172588,   26847297072,   2380855680;\n  9: 467832240, 9372384504, 82181624112, 412238074896, 1295476920549, 2618126589042, 3341678417748, 2502378994392, 903033995040, 71425670400;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(6*n^3+12*n^2+6*n)*x+(9*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(6*n^2+6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*v[i - 1])/(n)); 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-2feb6dc01f424b0a","steps":["basis A013610","D+1"],"edges":["e-45b4054949a52789","e-04e199e1724576a9"],"text":"A145324 \u2192 basis A013610 \u2192 D+1"},"headline":"A145324 \u2192 basis A013610 \u2192 D+1","chain":null}