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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":"b18f6b35e82c5a86dde09976be77fcc1bb2e71692f3bc75447e1c8def011138f","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-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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,    4;\n  2: 1,   20,     36;\n  3: 1,   56,    441,      576;\n  4: 1,  120,   2457,    13120,      14400;\n  5: 1,  220,   9207,   122320,     526900,      518400;\n  6: 1,  364,  27027,   711568,    7407400,    27832896,     25401600;\n  7: 1,  560,  67158,  3065920,   61886825,   550499040,   1881701136,    1625702400;\n  8: 1,  816, 147798, 10707008,  368478825,  6253988832,  49836284176,  158920556544,   131681894400;\n  9: 1, 1140, 296514, 31989920, 1723584525, 49233358980, 739338552904, 5431396008960, 16423522073856, 13168189440000;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(4*n^4-8*n^3+4*n^2)*x-(9*n^6-36*n^5+54*n^4-36*n^3+9*n^2)*x^2)*P(n-2,x) + ((4*n^4-8*n^3+4*n^2)*x^2+(4*n^6-16*n^5+24*n^4-16*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2-2*n+1+(4*n^4-8*n^3+4*n^2)*x)*P(n-1,x) - (4*n^4-8*n^3+4*n^2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (4*n^4 - 8*n^3 + 4*n^2)*x - (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*P[n - 1, x] - (4*n^4 - 8*n^3 + 4*n^2)*x^2*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (4*n^4 - 8*n^3 + 4*n^2)*x - (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*v[i - 2] + ((4*n^4 - 8*n^3 + 4*n^2)*x^2 + (4*n^6 - 16*n^5 + 24*n^4 - 16*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 2*n + 1 + (4*n^4 - 8*n^3 + 4*n^2)*x)*v[i - 1] - (4*n^4 - 8*n^3 + 4*n^2)*x^2*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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-e734a631f8279507","steps":["(\u03b8+1)\u00b2"],"edges":["e-cda364eb5d275d1a"],"text":"A008955 \u2192 (\u03b8+1)\u00b2"},"headline":"A008955 \u2192 (\u03b8+1)\u00b2","chain":null}