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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":"0b9b1253e5b9b69fb608d37aa7fac378d7f96e16fc159c1f579747bb535ff74b","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-6aadffa116a44288","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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,       24;\n  2:     12,      168,       243;\n  3:     48,      960,      2997,      1728;\n  4:    192,     4992,     23976,     32832,      10125;\n  5:    768,    24576,    159408,    352512,     273375,      52488;\n  6:   3072,   116736,    953856,   2930688,    3798900,    1907064,     250047;\n  7:  12288,   540672,   5329152,  21012480,   38685600,   33312384,   11752209,    1119744;\n  8:  49152,  2457600,  28366848, 136691712,  327888000,  408146688,  252047376,   66064896,   4782969;\n  9: 196608, 11010048, 145649664, 829882368, 2459808000, 4040036352, 3658973472, 1706489856, 345968091, 19683000;\n","recurrence_sentence":"The row polynomials satisfy -(16*n^3-32*n^2+20*n-4+(72*n^4-172*n^3+88*n^2+2*n-6)*x+(36*n^5-120*n^4-27*n^3+57*n^2-3*n-9)*x^2-(36*n^5+36*n^4+9*n^3)*x^3)*P(n,x) = -(64*n^3-128*n^2+80*n-16+(288*n^4-112*n^3-736*n^2+664*n-152)*x+(144*n^5+600*n^4-1572*n^3-966*n^2+1668*n-456)*x^2+(180*n^5-756*n^4-2007*n^3+1098*n-342)*x^3-(324*n^5+1080*n^4+837*n^3-135*n^2-135*n-27)*x^4)*P(n-1,x) + ((32*n^2-32*n+8)*x+(144*n^3+112*n^2-256*n+76)*x^2+(72*n^4+480*n^3-186*n^2-678*n+210)*x^3+(144*n^4-144*n^3-1044*n^2-720*n+144)*x^4-(216*n^4+864*n^3+1026*n^2+378*n+54)*x^5)*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[(-(64*n^3 - 128*n^2 + 80*n - 16 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (144*n^5 + 600*n^4 - 1572*n^3 - 966*n^2 + 1668*n - 456)*x^2 + (180*n^5 - 756*n^4 - 2007*n^3 + 1098*n - 342)*x^3 - (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^4)*P[n - 1, x] + ((32*n^2 - 32*n + 8)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (72*n^4 + 480*n^3 - 186*n^2 - 678*n + 210)*x^3 + (144*n^4 - 144*n^3 - 1044*n^2 - 720*n + 144)*x^4 - (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^5)*D[P[n - 1, x], x])/(-16*n^3 + 32*n^2 - 20*n + 4 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (36*n^5 - 120*n^4 - 27*n^3 + 57*n^2 - 3*n - 9)*x^2 + (36*n^5 + 36*n^4 + 9*n^3)*x^3)]];\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] = (-(64*n^3 - 128*n^2 + 80*n - 16 + (288*n^4 - 112*n^3 - 736*n^2 + 664*n - 152)*x + (144*n^5 + 600*n^4 - 1572*n^3 - 966*n^2 + 1668*n - 456)*x^2 + (180*n^5 - 756*n^4 - 2007*n^3 + 1098*n - 342)*x^3 - (324*n^5 + 1080*n^4 + 837*n^3 - 135*n^2 - 135*n - 27)*x^4)*v[i - 1] + ((32*n^2 - 32*n + 8)*x + (144*n^3 + 112*n^2 - 256*n + 76)*x^2 + (72*n^4 + 480*n^3 - 186*n^2 - 678*n + 210)*x^3 + (144*n^4 - 144*n^3 - 1044*n^2 - 720*n + 144)*x^4 - (216*n^4 + 864*n^3 + 1026*n^2 + 378*n + 54)*x^5)*deriv(v[i - 1], x))/(-16*n^3 + 32*n^2 - 20*n + 4 - (72*n^4 - 172*n^3 + 88*n^2 + 2*n - 6)*x - (36*n^5 - 120*n^4 - 27*n^3 + 57*n^2 - 3*n - 9)*x^2 + (36*n^5 + 36*n^4 + 9*n^3)*x^3)); 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-c85abc56078671cc","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-12f73d6f0ea24cf5","e-72c7dfeb6a0cdcd8"],"text":"A232535 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A232535 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}