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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":"e47d4e17f1bd90ceec67970b8b411883aacc0759c3fc6515718cf9a73ee4a868","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-3d0ce3ba50a1aaca","role":"sequence","name":"basis_map_A013609 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:      4,      16;\n  2:     16,     128,      108;\n  3:     64,     768,     1440,       512;\n  4:    256,    4096,    12096,     10240,      2000;\n  5:   1024,   20480,    82944,    114688,     56000,      6912;\n  6:   4096,   98304,   506880,    983040,    806400,    258048,     21952;\n  7:  16384,  458752,  2875392,   7208960,   8448000,   4644864,   1053696,     65536;\n  8:  65536, 2097152, 15482880,  47710208,  73216000,  58392576,  23181312,   3932160,   186624;\n  9: 262144, 9437184, 80216064, 293601280, 559104000, 590413824, 344408064, 103809024, 13685760, 512000;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3+4*n^2+n+(12*n^4+6*n^3-2*n^2-n)*x+(4*n^5+4*n^4+n^3)*x^2)*P(n,x) = (16*n^3+16*n^2+4*n+(48*n^4+120*n^3+56*n^2-12*n-8)*x+(16*n^5+136*n^4+172*n^3-2*n^2-44*n-2)*x^2+(24*n^5+80*n^4+62*n^3-10*n^2-10*n-2)*x^3)*P(n-1,x) - ((8*n^2+8*n+2)*x+(24*n^3+64*n^2+40*n+6)*x^2+(8*n^4+80*n^3+134*n^2+62*n+6)*x^3+(16*n^4+64*n^3+76*n^2+28*n+4)*x^4)*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[((16*n^3 + 16*n^2 + 4*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (24*n^5 + 80*n^4 + 62*n^3 - 10*n^2 - 10*n - 2)*x^3)*P[n - 1, x] - ((8*n^2 + 8*n + 2)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (16*n^4 + 64*n^3 + 76*n^2 + 28*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^3 + 4*n^2 + n + (12*n^4 + 6*n^3 - 2*n^2 - n)*x + (4*n^5 + 4*n^4 + n^3)*x^2)]];\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] = ((16*n^3 + 16*n^2 + 4*n + (48*n^4 + 120*n^3 + 56*n^2 - 12*n - 8)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (24*n^5 + 80*n^4 + 62*n^3 - 10*n^2 - 10*n - 2)*x^3)*v[i - 1] - ((8*n^2 + 8*n + 2)*x + (24*n^3 + 64*n^2 + 40*n + 6)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (16*n^4 + 64*n^3 + 76*n^2 + 28*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^3 + 4*n^2 + n + (12*n^4 + 6*n^3 - 2*n^2 - n)*x + (4*n^5 + 4*n^4 + n^3)*x^2)); 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-2245b35ec7cd4857","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-11b86d48c22b376e","e-af3ee7a2219332ef"],"text":"A091044 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A091044 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}