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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":"2107992c7a9ee8abe43f968d755ee416f2166fc8bf768e663e7fddc0a87fcac6","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-3d55772447340781","role":"sequence","name":"basis_map_A007318 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,       8;\n  2:     16,      64,       27;\n  3:     64,     384,      360,       64;\n  4:    256,    2048,     3024,     1280,      125;\n  5:   1024,   10240,    20736,    14336,     3500,      216;\n  6:   4096,   49152,   126720,   122880,    50400,     8064,     343;\n  7:  16384,  229376,   718848,   901120,   528000,   145152,   16464,    512;\n  8:  65536, 1048576,  3870720,  5963776,  4576000,  1824768,  362208,  30720,   729;\n  9: 262144, 4718592, 20054016, 36700160, 34944000, 18450432, 5381376, 811008, 53460, 1000;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3+16*n^2+4*n+(24*n^4+12*n^3-4*n^2-2*n)*x+(4*n^5+4*n^4+n^3)*x^2)*P(n,x) = (64*n^3+64*n^2+16*n+(96*n^4+240*n^3+112*n^2-24*n-16)*x+(16*n^5+136*n^4+172*n^3-2*n^2-44*n-2)*x^2+(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^3)*P(n-1,x) - ((32*n^2+32*n+8)*x+(48*n^3+128*n^2+80*n+12)*x^2+(8*n^4+80*n^3+134*n^2+62*n+6)*x^3+(8*n^4+32*n^3+38*n^2+14*n+2)*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[((64*n^3 + 64*n^2 + 16*n + (96*n^4 + 240*n^3 + 112*n^2 - 24*n - 16)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*P[n - 1, x] - ((32*n^2 + 32*n + 8)*x + (48*n^3 + 128*n^2 + 80*n + 12)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*D[P[n - 1, x], x])/(16*n^3 + 16*n^2 + 4*n + (24*n^4 + 12*n^3 - 4*n^2 - 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] = ((64*n^3 + 64*n^2 + 16*n + (96*n^4 + 240*n^3 + 112*n^2 - 24*n - 16)*x + (16*n^5 + 136*n^4 + 172*n^3 - 2*n^2 - 44*n - 2)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*v[i - 1] - ((32*n^2 + 32*n + 8)*x + (48*n^3 + 128*n^2 + 80*n + 12)*x^2 + (8*n^4 + 80*n^3 + 134*n^2 + 62*n + 6)*x^3 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*deriv(v[i - 1], x))/(16*n^3 + 16*n^2 + 4*n + (24*n^4 + 12*n^3 - 4*n^2 - 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 A007318","(\u03b8+1)\u00b2"],"edges":["e-13cc0c6e042058d4","e-b059608fafce0522"],"text":"A091044 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A091044 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}