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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":"680e4abce441b57d41eaeb3a5a8e42916bedfbf76fbf9e6acb7cef82642693c5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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:      4,       36;\n   3:     16,      240,       405;\n   4:     64,     1344,      4536,       3024;\n   5:    256,     6912,     34992,      51840,      18225;\n   6:   1024,    33792,    228096,     532224,     445500,      96228;\n   7:   4096,   159744,   1347840,    4313088,    5896800,    3184272,     464373;\n   8:  16384,   737280,   7464960,   30412800,   58320000,   52907904,   20003760,    2099520;\n   9:  65536,  3342336,  39481344,  195526656,  484704000,  628176384,  408076704,  114213888,   9034497;\n  10: 262144, 14942208, 201719808, 1176698880, 3585254400, 6084688896, 5733649152, 2808317952, 605842740, 37397700;\n","recurrence_sentence":"The row polynomials satisfy (32*n^3-112*n^2+128*n-48+(144*n^4-648*n^3+1068*n^2-762*n+198)*x+(72*n^5-324*n^4+558*n^3-459*n^2+180*n-27)*x^2)*P(n,x) = (128*n^3-448*n^2+512*n-192+(576*n^4-1440*n^3+240*n^2+1560*n-936)*x+(288*n^5+864*n^4-4824*n^3+4896*n^2-72*n-1188)*x^2+(648*n^5-1188*n^4-594*n^3+1701*n^2-270*n-324)*x^3)*P(n-1,x) - ((64*n^2-128*n+64)*x+(288*n^3-240*n^2-360*n+324)*x^2+(144*n^4+864*n^3-1548*n^2+144*n+450)*x^3+(432*n^4-756*n^2+216*n+162)*x^4)*d/dx 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[((128*n^3 - 448*n^2 + 512*n - 192 + (576*n^4 - 1440*n^3 + 240*n^2 + 1560*n - 936)*x + (288*n^5 + 864*n^4 - 4824*n^3 + 4896*n^2 - 72*n - 1188)*x^2 + (648*n^5 - 1188*n^4 - 594*n^3 + 1701*n^2 - 270*n - 324)*x^3)*P[n - 1, x] - ((64*n^2 - 128*n + 64)*x + (288*n^3 - 240*n^2 - 360*n + 324)*x^2 + (144*n^4 + 864*n^3 - 1548*n^2 + 144*n + 450)*x^3 + (432*n^4 - 756*n^2 + 216*n + 162)*x^4)*D[P[n - 1, x], x])/(32*n^3 - 112*n^2 + 128*n - 48 + (144*n^4 - 648*n^3 + 1068*n^2 - 762*n + 198)*x + (72*n^5 - 324*n^4 + 558*n^3 - 459*n^2 + 180*n - 27)*x^2)]];\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] = ((128*n^3 - 448*n^2 + 512*n - 192 + (576*n^4 - 1440*n^3 + 240*n^2 + 1560*n - 936)*x + (288*n^5 + 864*n^4 - 4824*n^3 + 4896*n^2 - 72*n - 1188)*x^2 + (648*n^5 - 1188*n^4 - 594*n^3 + 1701*n^2 - 270*n - 324)*x^3)*v[i - 1] - ((64*n^2 - 128*n + 64)*x + (288*n^3 - 240*n^2 - 360*n + 324)*x^2 + (144*n^4 + 864*n^3 - 1548*n^2 + 144*n + 450)*x^3 + (432*n^4 - 756*n^2 + 216*n + 162)*x^4)*deriv(v[i - 1], x))/(32*n^3 - 112*n^2 + 128*n - 48 + (144*n^4 - 648*n^3 + 1068*n^2 - 762*n + 198)*x + (72*n^5 - 324*n^4 + 558*n^3 - 459*n^2 + 180*n - 27)*x^2)); 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-ba8de4e02acfb6ca","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-4aa0aeb00b1923c9","e-527ff639ed8d8dcc"],"text":"A091042 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A091042 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}