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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":"d592c1db608f0b7034d7c463dc78705b354efb27089a198552fedff1352f11b7","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-f8c44f01f6cd4a34","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+3x).","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,      12;\n  2:     12,      84,      108;\n  3:     45,     468,     1188,       864;\n  4:    171,    2340,     8856,     12960,      6480;\n  5:    648,   11016,    55404,    121824,    123120,      46656;\n  6:   2457,   49896,   312984,    918432,   1399680,    1073088,     326592;\n  7:   9315,  219996,  1653372,   6067872,  12370320,   14323392,    8817984,    2239488;\n  8:  35316,  950940,  8328096,  36689760,  93707280,  145333440,  135209088,   69424128,  15116544;\n  9: 133893, 4048380, 40482828, 208124640, 638915040, 1242729216, 1549679040, 1202605056, 529079040, 100776960;\n","recurrence_sentence":"The row polynomials satisfy (28*n+(28*n^2+76*n)*x+(8*n^3+48*n^2+40*n)*x^2)*P(n,x) = -(56*n-140+(84*n^2-100*n-632)*x+(16*n^3+56*n^2-664*n-944)*x^2-(96*n^2+576*n+480)*x^3)*P(n-1,x) + (49+(98*n+308)*x+(28*n^2+448*n+704)*x^2+(80*n^2+648*n+688)*x^3+(48*n^2+288*n+240)*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[(-(56*n - 140 + (84*n^2 - 100*n - 632)*x + (16*n^3 + 56*n^2 - 664*n - 944)*x^2 - (96*n^2 + 576*n + 480)*x^3)*P[n - 1, x] + (49 + (98*n + 308)*x + (28*n^2 + 448*n + 704)*x^2 + (80*n^2 + 648*n + 688)*x^3 + (48*n^2 + 288*n + 240)*x^4)*D[P[n - 1, x], x])/(28*n + (28*n^2 + 76*n)*x + (8*n^3 + 48*n^2 + 40*n)*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] = (-(56*n - 140 + (84*n^2 - 100*n - 632)*x + (16*n^3 + 56*n^2 - 664*n - 944)*x^2 - (96*n^2 + 576*n + 480)*x^3)*v[i - 1] + (49 + (98*n + 308)*x + (28*n^2 + 448*n + 704)*x^2 + (80*n^2 + 648*n + 688)*x^3 + (48*n^2 + 288*n + 240)*x^4)*deriv(v[i - 1], x))/(28*n + (28*n^2 + 76*n)*x + (8*n^3 + 48*n^2 + 40*n)*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-016d7724d51f6416","steps":["basis A013609","\u03b8+1"],"edges":["e-2339e629d7f9e0b9","e-7d1efee71aa99117"],"text":"A099093 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A099093 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}