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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":"4f402286f4a0915972cba4a61709bda881088dcf56f423c1e031d50cc3806605","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-84c761192e0db3c0","role":"sequence","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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,    16,     48;\n  1:   2,    40,    192,     256;\n  2:   4,    96,    624,    1536,     1280;\n  3:   8,   224,   1824,    6400,    10240,     6144;\n  4:  16,   512,   4992,   22528,    52480,    61440,     28672;\n  5:  32,  1152,  13056,   71680,   217600,   374784,    344064,    131072;\n  6:  64,  2560,  33024,  212992,   793600,  1794048,   2437120,   1835008,    589824;\n  7: 128,  5632,  81408,  602112,  2652160,  7397376,  13246464,  14811136,   9437184,   2621440;\n  8: 256, 12288, 196608, 1638400,  8314880, 27525120,  61014016,  90177536,  85524480,  47185920,  11534336;\n  9: 512, 26624, 466944, 4325376, 24821760, 94961664, 250478592, 459276288, 576847872, 474480640, 230686720, 50331648;\n","recurrence_sentence":"The row polynomials satisfy (1+(2*n+12)*x+(8*n+16)*x^2)*P(n,x) = (2+(4*n+32)*x+(24*n+104)*x^2+(32*n+96)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 16*x + 48*x^2.","mathematica":"P[0, x_] := 1 + 16*x + 48*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + (4*n + 32)*x + (24*n + 104)*x^2 + (32*n + 96)*x^3)*P[n - 1, x])/(1 + (2*n + 12)*x + (8*n + 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 16*x + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + (4*n + 32)*x + (24*n + 104)*x^2 + (32*n + 96)*x^3)*v[i - 1])/(1 + (2*n + 12)*x + (8*n + 16)*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-7717563017cb9e5f","steps":["basis A013611","\u03b8+1"],"edges":["e-d3bf3843e93b142d","e-e4feb0e7b7d63679"],"text":"A159854 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A159854 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}