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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":"32c055cb4ba656bc68d86719dbb10c06eefcf367d6991e346a032f6aa6cec238","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-44370b381098ad54","role":"sequence","name":"basis_map_A013610 of A Riordan array of coordination sequences.","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,     6;\n  2:    7,    36,     27;\n  3:   17,   138,    243,     108;\n  4:   41,   456,   1296,    1296,     405;\n  5:   99,  1398,   5508,    8856,    6075,     1458;\n  6:  239,  4092,  20655,   46224,   50625,    26244,    5103;\n  7:  577, 11610,  71523,  205956,  313875,   258066,  107163,   17496;\n  8: 1393, 32208, 234360,  826848, 1624050,  1854576, 1214514,  419904,   59049;\n  9: 3363, 87846, 737424, 3083184, 7434990, 10943748, 9930438, 5388768, 1594323, 196830;\n","recurrence_sentence":"The row polynomials satisfy (96+(360*n+288)*x+(432*n^2+540*n+270)*x^2+(162*n^3+324*n^2-162*n)*x^3-(243*n^2+243*n)*x^4)*P(n,x) = -(192*n-480+(432*n^2-360*n-2592)*x+(216*n^3+648*n^2-4320*n-4158)*x^2+(486*n^3-1944*n^2-5994*n-972)*x^3-(2187*n^2+729*n-2430)*x^4+(1458*n+1458)*x^5)*P(n-1,x) + (64+(192*n+480)*x+(144*n^2+1296*n+1080)*x^2+(864*n^2+2592*n+108)*x^3+(1458*n^2+810*n-2430)*x^4+(486*n^2-1701*n-2673)*x^5-(729*n+729)*x^6)*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[(-(192*n - 480 + (432*n^2 - 360*n - 2592)*x + (216*n^3 + 648*n^2 - 4320*n - 4158)*x^2 + (486*n^3 - 1944*n^2 - 5994*n - 972)*x^3 - (2187*n^2 + 729*n - 2430)*x^4 + (1458*n + 1458)*x^5)*P[n - 1, x] + (64 + (192*n + 480)*x + (144*n^2 + 1296*n + 1080)*x^2 + (864*n^2 + 2592*n + 108)*x^3 + (1458*n^2 + 810*n - 2430)*x^4 + (486*n^2 - 1701*n - 2673)*x^5 - (729*n + 729)*x^6)*D[P[n - 1, x], x])/(96 + (360*n + 288)*x + (432*n^2 + 540*n + 270)*x^2 + (162*n^3 + 324*n^2 - 162*n)*x^3 - (243*n^2 + 243*n)*x^4)]];\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] = (-(192*n - 480 + (432*n^2 - 360*n - 2592)*x + (216*n^3 + 648*n^2 - 4320*n - 4158)*x^2 + (486*n^3 - 1944*n^2 - 5994*n - 972)*x^3 - (2187*n^2 + 729*n - 2430)*x^4 + (1458*n + 1458)*x^5)*v[i - 1] + (64 + (192*n + 480)*x + (144*n^2 + 1296*n + 1080)*x^2 + (864*n^2 + 2592*n + 108)*x^3 + (1458*n^2 + 810*n - 2430)*x^4 + (486*n^2 - 1701*n - 2673)*x^5 - (729*n + 729)*x^6)*deriv(v[i - 1], x))/(96 + (360*n + 288)*x + (432*n^2 + 540*n + 270)*x^2 + (162*n^3 + 324*n^2 - 162*n)*x^3 - (243*n^2 + 243*n)*x^4)); 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-ec35d1145e9a1211","steps":["basis A013610","\u03b8+1"],"edges":["e-3d8a64e4b99d997d","e-4071f685cf3f6c6b"],"text":"A113413 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A113413 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}