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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"510353917d938c25f3ac8d22a8f5dd9a4f1422046a99bbb43fb35cd6903d0108","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-11b174a1f5e78a68","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","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]"}]},{"id":"e-43c85baf334c8872","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-3d28544a90bbf61e","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a 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":"4126d44608550dc4781dcfefd94d382317a3379560c7913a35654836fe5b23b4","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-07e2f0a7469aac14","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","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: 0, 0, 3, 4;\n   3: 0, 0, 0, 0, 5, 12, 7;\n   4: 0, 0, 0, 0, 0,  0, 7, 24, 27, 10;\n   5: 0, 0, 0, 0, 0,  0, 0,  0,  9, 40, 66, 48,  13;\n   6: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 11, 60, 130, 140,  75,  16;\n   7: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  13,  84, 225, 320, 255, 108,  19;\n   8: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  15, 112, 357, 630, 665,  420,  147,   22;\n   9: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  17, 144, 532, 1120, 1470, 1232,  644,  192,   25;\n  10: 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  19,  180,  756, 1848, 2898, 3024, 2100, 936, 243, 28;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3+(3*n-5)*x)*P(n,x) = ((2*n-1)*x^2+(5*n-3)*x^3+(3*n-2)*x^4)*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[(((2*n - 1)*x^2 + (5*n - 3)*x^3 + (3*n - 2)*x^4)*P[n - 1, x])/(2*n - 3 + (3*n - 5)*x)]];\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] = (((2*n - 1)*x^2 + (5*n - 3)*x^3 + (3*n - 2)*x^4)*v[i - 1])/(2*n - 3 + (3*n - 5)*x)); 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-11b174a1f5e78a68","steps":["\u03b8+1"],"edges":["e-351367db9c8908f5"],"text":"Veronese parent of A098172 \u2192 \u03b8+1"},"headline":"Veronese parent of A098172 \u2192 \u03b8+1","chain":null}