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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":"73179ff93ec6ec5c15ac3b08c4eb512853695d9cb582c790d91f373ec6ef6fda","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-a19ad5e2037d7c6a","role":"sequence","name":"basis_map_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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:        6,        28,         18;\n  2:       36,       336,        657,         448,         100;\n  3:      216,      3024,       9882,       13552,        9150,         3024,          392;\n  4:     1296,     24192,     110808,      228480,      258025,       171360,        67032,        14336,         1296;\n  5:     7776,    181440,    1069200,     2943360,     4662750,      4657212,      3046330,      1308160,       356400,        56000,         3872;\n  6:    46656,   1306368,    9412848,    32417280,    66109500,     88188912,     80928841,     52260096,     23799420,      7504000,      1562352,       193536,       10816;\n  7:   279936,   9144576,   77892192,   322092288,   803514600,   1341394992,   1584966642,   1367041984,    873348966,    414010800,    144037432,     35788032,     6019104,      614656,      28800;\n  8:  1679616,  62705664,  616419072,  2975035392,  8776706400,  17597212416,  25440090480,  27462779904,  22591325169,  14303531200,   6980160880,   2606994432,   732475744,   149976064,   21139200,   1835008,    73984;\n  9: 10077696, 423263232, 4716361728, 26022850560, 88624238400, 207440134272, 354707561376, 459939658752, 462959070966, 366876048700, 230526944802, 114984914688, 45310338528, 13943164032, 3282379200, 571146240, 69249024, 5225472, 184832;\n","recurrence_sentence":"The row polynomials satisfy (36+(126*n-42)*x+(49*n^2+96*n-72)*x^2+(56*n^2-14*n-14)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (216+(756*n+756)*x+(294*n^2+2046*n+1098)*x^2+(679*n^2+2198*n+847)*x^3+(586*n^2+1170*n+366)*x^4+(224*n^2+308*n+84)*x^5+(32*n^2+32*n+8)*x^6)*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[((216 + (756*n + 756)*x + (294*n^2 + 2046*n + 1098)*x^2 + (679*n^2 + 2198*n + 847)*x^3 + (586*n^2 + 1170*n + 366)*x^4 + (224*n^2 + 308*n + 84)*x^5 + (32*n^2 + 32*n + 8)*x^6)*P[n - 1, x])/(36 + (126*n - 42)*x + (49*n^2 + 96*n - 72)*x^2 + (56*n^2 - 14*n - 14)*x^3 + (16*n^2 - 16*n + 4)*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] = ((216 + (756*n + 756)*x + (294*n^2 + 2046*n + 1098)*x^2 + (679*n^2 + 2198*n + 847)*x^3 + (586*n^2 + 1170*n + 366)*x^4 + (224*n^2 + 308*n + 84)*x^5 + (32*n^2 + 32*n + 8)*x^6)*v[i - 1])/(36 + (126*n - 42)*x + (49*n^2 + 96*n - 72)*x^2 + (56*n^2 - 14*n - 14)*x^3 + (16*n^2 - 16*n + 4)*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-eed4f744ce063213","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-6e97878f95ac8f6d","e-6bc952b1f59f15a4"],"text":"A200536 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A200536 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}