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(f(k))_(k >= 0) is a multiplier sequence, so x^k -> 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":"a529ab627c0cbbf8a1bd19e8ec900a1efec82f8e46ee038c9dccd09e6f98a687","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-0c0e99d973615582","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.","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,       12;\n  2:     9,      120,         81;\n  3:    27,      852,       1917,         432;\n  4:    81,     5232,      26838,       20928,        2025;\n  5:   243,    29724,     293814,      522336,      185775,         8748;\n  6:   729,   161256,    2792367,     9402048,     7756575,      1451304,       35721;\n  7:  2187,   850020,   24263307,   139224720,   217538625,     97053228,    10412745,      139968;\n  8:  6561,  4398816,  198531324,  1813774464,  4785849750,   4080992544,  1080892764,    70381056,    531441;\n  9: 19683, 22492716, 1557973980, 21618138816, 89486641050, 128860763112, 66205550124, 11078926080, 455477499, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+4*n+1)*P(n,x) = -(12*n^2+24*n+9+(96*n^3+144*n^2+8*n-52)*x-(144*n^4+256*n^3+56*n^2-64*n-23)*x^2)*P(n-2,x) - ((48*n^2+96*n+56)*x-(96*n^3+304*n^2+312*n+132)*x^2-(64*n^4-96*n^3-448*n^2-344*n-76)*x^3)*d/dx P(n-2,x) - ((16*n^2+32*n+20)*x^2-(64*n^3+160*n^2+128*n+40)*x^3+(64*n^3+144*n^2+96*n+20)*x^4)*d^2/dx^2 P(n-2,x) + (16*n^2+20*n+6+(32*n^3+48*n^2+24*n+4)*x)*P(n-1,x) + ((16*n^2+24*n+12)*x-(32*n^3+80*n^2+56*n+12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n^2 + 24*n + 9 + (96*n^3 + 144*n^2 + 8*n - 52)*x - (144*n^4 + 256*n^3 + 56*n^2 - 64*n - 23)*x^2)*P[n - 2, x] - ((48*n^2 + 96*n + 56)*x - (96*n^3 + 304*n^2 + 312*n + 132)*x^2 - (64*n^4 - 96*n^3 - 448*n^2 - 344*n - 76)*x^3)*D[P[n - 2, x], x] - ((16*n^2 + 32*n + 20)*x^2 - (64*n^3 + 160*n^2 + 128*n + 40)*x^3 + (64*n^3 + 144*n^2 + 96*n + 20)*x^4)*D[P[n - 2, x], {x, 2}] + (16*n^2 + 20*n + 6 + (32*n^3 + 48*n^2 + 24*n + 4)*x)*P[n - 1, x] + ((16*n^2 + 24*n + 12)*x - (32*n^3 + 80*n^2 + 56*n + 12)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 4*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^2 + 24*n + 9 + (96*n^3 + 144*n^2 + 8*n - 52)*x - (144*n^4 + 256*n^3 + 56*n^2 - 64*n - 23)*x^2)*v[i - 2] - ((48*n^2 + 96*n + 56)*x - (96*n^3 + 304*n^2 + 312*n + 132)*x^2 - (64*n^4 - 96*n^3 - 448*n^2 - 344*n - 76)*x^3)*deriv(v[i - 2], x) - ((16*n^2 + 32*n + 20)*x^2 - (64*n^3 + 160*n^2 + 128*n + 40)*x^3 + (64*n^3 + 144*n^2 + 96*n + 20)*x^4)*deriv(deriv(v[i - 2], x), x) + (16*n^2 + 20*n + 6 + (32*n^3 + 48*n^2 + 24*n + 4)*x)*v[i - 1] + ((16*n^2 + 24*n + 12)*x - (32*n^3 + 80*n^2 + 56*n + 12)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 4*n + 1)); 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-0c0e99d973615582","steps":["(\u03b8+1)\u00b2"],"edges":["e-2e186ccd88e6aa08"],"text":"A257611 \u2192 (\u03b8+1)\u00b2"},"headline":"A257611 \u2192 (\u03b8+1)\u00b2","chain":null}