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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":"8f874452a4ce4ee1effe4c78a5c784e3e6b5d8566326293410a9a2eaf5293f79","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-14e332ad5b6d2ede","role":"sequence","name":"Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.","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: 1,  6,    3;\n  2: 1, 14,   39,    28,      5;\n  3: 1, 22,  123,   252,    205,     66,       7;\n  4: 1, 30,  255,   924,   1605,   1386,     595,     120,        9;\n  5: 1, 38,  435,  2300,   6445,  10098,    9023,    4600,     1305,      190,       11;\n  6: 1, 46,  663,  4636,  18245,  43098,   62923,   57464,    32841,    11590,     2431,     276,      13;\n  7: 1, 54,  939,  8188,  41805, 134178,  280567,  389112,   360729,   223630,    91971,   24564,    4069,     378,     15;\n  8: 1, 62, 1263, 13212,  83205, 340170,  939715, 1793144,  2391561,  2241430,  1476695,  680340,  216333,   46242,   6315,   496,   17;\n  9: 1, 70, 1635, 19964, 149805, 747090, 2585135, 6363640, 11308185, 14625630, 13821115, 9545460, 4800965, 1743210, 449415, 79856, 9265, 630, 19;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^2-2*n+(16*n^3-24*n^2+8*n)*x-(16*n^3-4*n^2-2*n)*x^2)*P(n,x) = -(4*n^2-2*n+(16*n^3+24*n^2-28*n)*x+(80*n^3-12*n^2-82*n+10)*x^2+(32*n^3-48*n^2-44*n)*x^3-(16*n^2+12*n+2)*x^4)*P(n-1,x) + (2*n*x+(8*n^2+20*n-1)*x^2+(56*n^2+56*n-5)*x^3+(56*n^2+44*n+5)*x^4+(8*n^2+6*n+1)*x^5)*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[(-(4*n^2 - 2*n + (16*n^3 + 24*n^2 - 28*n)*x + (80*n^3 - 12*n^2 - 82*n + 10)*x^2 + (32*n^3 - 48*n^2 - 44*n)*x^3 - (16*n^2 + 12*n + 2)*x^4)*P[n - 1, x] + (2*n*x + (8*n^2 + 20*n - 1)*x^2 + (56*n^2 + 56*n - 5)*x^3 + (56*n^2 + 44*n + 5)*x^4 + (8*n^2 + 6*n + 1)*x^5)*D[P[n - 1, x], x])/(-4*n^2 + 2*n - (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 4*n^2 - 2*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] = (-(4*n^2 - 2*n + (16*n^3 + 24*n^2 - 28*n)*x + (80*n^3 - 12*n^2 - 82*n + 10)*x^2 + (32*n^3 - 48*n^2 - 44*n)*x^3 - (16*n^2 + 12*n + 2)*x^4)*v[i - 1] + (2*n*x + (8*n^2 + 20*n - 1)*x^2 + (56*n^2 + 56*n - 5)*x^3 + (56*n^2 + 44*n + 5)*x^4 + (8*n^2 + 6*n + 1)*x^5)*deriv(v[i - 1], x))/(-4*n^2 + 2*n - (16*n^3 - 24*n^2 + 8*n)*x + (16*n^3 - 4*n^2 - 2*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-cdb0db4ad473ee28","steps":["basis A008288","\u03b8+1"],"edges":["e-054f97e0823e3ab6","e-6c6d3e3fee2faca8"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008288 \u2192 \u03b8+1","chain":null}