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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"302a19f7dadc229b8fb2f23555e67fc0b6085db37f9f524e90b1825bfa965f56","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-0a5e54eacdb24124","role":"sequence","name":"Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 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: 0,     2;\n  2: 0,     6,      6;\n  3: 0,    20,     36,     12;\n  4: 0,    70,    180,    120,     20;\n  5: 0,   252,    840,    840,    300,     30;\n  6: 0,   924,   3780,   5040,   2800,    630,     42;\n  7: 0,  3432,  16632,  27720,  21000,   7560,   1176,    56;\n  8: 0, 12870,  72072, 144144, 138600,  69300,  17640,  2016,   72;\n  9: 0, 48620, 308880, 720720, 840840, 540540, 194040, 36960, 3240, 90;\n","recurrence_sentence":"The row polynomials satisfy (8*n^5-64*n^4+190*n^3-248*n^2+120*n)*P(n,x) = -((96*n^5-912*n^4+3336*n^3-5868*n^2+4968*n-1620)*x^2-(8*n^5-76*n^4+262*n^3-383*n^2+189*n+18)*x^3)*P(n-3,x) - ((32*n^5-304*n^4+1112*n^3-1956*n^2+1656*n-540)*x^3+(6*n^3-39*n^2+81*n-54)*x^4)*d/dx P(n-3,x) - (128*n^5-1152*n^4+4032*n^3-6848*n^2+5640*n-1800-(32*n^5-288*n^4+992*n^3-1608*n^2+1190*n-300)*x+(24*n^5-216*n^4+750*n^3-1236*n^2+936*n-240)*x^2)*P(n-2,x) + ((128*n^5-1152*n^4+4032*n^3-6848*n^2+5640*n-1800)*x+(64*n^5-576*n^4+2056*n^3-3648*n^2+3226*n-1140)*x^2+(12*n^3-72*n^2+144*n-96)*x^3)*d/dx P(n-2,x) + (64*n^5-544*n^4+1776*n^3-2760*n^2+2016*n-540+(24*n^5-204*n^4+678*n^3-1101*n^2+873*n-270)*x)*P(n-1,x) - ((32*n^5-272*n^4+888*n^3-1388*n^2+1040*n-300)*x+(6*n^3-33*n^2+57*n-30)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2*x, P(2,x) = 6*x + 6*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[2, x_] := 6*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-((96*n^5 - 912*n^4 + 3336*n^3 - 5868*n^2 + 4968*n - 1620)*x^2 - (8*n^5 - 76*n^4 + 262*n^3 - 383*n^2 + 189*n + 18)*x^3)*P[n - 3, x] - ((32*n^5 - 304*n^4 + 1112*n^3 - 1956*n^2 + 1656*n - 540)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*D[P[n - 3, x], x] - (128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800 - (32*n^5 - 288*n^4 + 992*n^3 - 1608*n^2 + 1190*n - 300)*x + (24*n^5 - 216*n^4 + 750*n^3 - 1236*n^2 + 936*n - 240)*x^2)*P[n - 2, x] + ((128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800)*x + (64*n^5 - 576*n^4 + 2056*n^3 - 3648*n^2 + 3226*n - 1140)*x^2 + (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*D[P[n - 2, x], x] + (64*n^5 - 544*n^4 + 1776*n^3 - 2760*n^2 + 2016*n - 540 + (24*n^5 - 204*n^4 + 678*n^3 - 1101*n^2 + 873*n - 270)*x)*P[n - 1, x] - ((32*n^5 - 272*n^4 + 888*n^3 - 1388*n^2 + 1040*n - 300)*x + (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*D[P[n - 1, x], x])/(8*n^5 - 64*n^4 + 190*n^3 - 248*n^2 + 120*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; v[3] = 6*x + 6*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-((96*n^5 - 912*n^4 + 3336*n^3 - 5868*n^2 + 4968*n - 1620)*x^2 - (8*n^5 - 76*n^4 + 262*n^3 - 383*n^2 + 189*n + 18)*x^3)*v[i - 3] - ((32*n^5 - 304*n^4 + 1112*n^3 - 1956*n^2 + 1656*n - 540)*x^3 + (6*n^3 - 39*n^2 + 81*n - 54)*x^4)*deriv(v[i - 3], x) - (128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800 - (32*n^5 - 288*n^4 + 992*n^3 - 1608*n^2 + 1190*n - 300)*x + (24*n^5 - 216*n^4 + 750*n^3 - 1236*n^2 + 936*n - 240)*x^2)*v[i - 2] + ((128*n^5 - 1152*n^4 + 4032*n^3 - 6848*n^2 + 5640*n - 1800)*x + (64*n^5 - 576*n^4 + 2056*n^3 - 3648*n^2 + 3226*n - 1140)*x^2 + (12*n^3 - 72*n^2 + 144*n - 96)*x^3)*deriv(v[i - 2], x) + (64*n^5 - 544*n^4 + 1776*n^3 - 2760*n^2 + 2016*n - 540 + (24*n^5 - 204*n^4 + 678*n^3 - 1101*n^2 + 873*n - 270)*x)*v[i - 1] - ((32*n^5 - 272*n^4 + 888*n^3 - 1388*n^2 + 1040*n - 300)*x + (6*n^3 - 33*n^2 + 57*n - 30)*x^2)*deriv(v[i - 1], x))/(8*n^5 - 64*n^4 + 190*n^3 - 248*n^2 + 120*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-0a5e54eacdb24124","steps":["\u03b8+1"],"edges":["e-1895a08712ed0f37"],"text":"A360282 \u2192 \u03b8+1"},"headline":"A360282 \u2192 \u03b8+1","chain":null}