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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":"0b277203ccc892ba34da49ed9e3519803316d7578da7a748096132eadac1b292","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-e189038e95d51606","role":"sequence","name":"Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, 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;\n  2: 1, 12;\n  3: 1, 24;\n  4: 1, 36,   81;\n  5: 1, 48,  243;\n  6: 1, 60,  486,  432;\n  7: 1, 72,  810, 1728;\n  8: 1, 84, 1215, 4320,  2025;\n  9: 1, 96, 1701, 8640, 10125;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n+(9*n^3-6*n)*x+(9*n^4+9*n^3)*x^2)*P(n,x) = (n^2+n+(9*n^3+24*n^2-6*n-24)*x+(9*n^4+99*n^3+72*n^2-126*n-36)*x^2+(54*n^4+216*n^3+162*n^2-216*n-216)*x^3)*P(n-1,x) - ((n+1)*x+(9*n^2+30*n+18)*x^2+(9*n^3+153*n^2+288*n+108)*x^3+(108*n^3+540*n^2+864*n+432)*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[((n^2 + n + (9*n^3 + 24*n^2 - 6*n - 24)*x + (9*n^4 + 99*n^3 + 72*n^2 - 126*n - 36)*x^2 + (54*n^4 + 216*n^3 + 162*n^2 - 216*n - 216)*x^3)*P[n - 1, x] - ((n + 1)*x + (9*n^2 + 30*n + 18)*x^2 + (9*n^3 + 153*n^2 + 288*n + 108)*x^3 + (108*n^3 + 540*n^2 + 864*n + 432)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (9*n^3 - 6*n)*x + (9*n^4 + 9*n^3)*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] = ((n^2 + n + (9*n^3 + 24*n^2 - 6*n - 24)*x + (9*n^4 + 99*n^3 + 72*n^2 - 126*n - 36)*x^2 + (54*n^4 + 216*n^3 + 162*n^2 - 216*n - 216)*x^3)*v[i - 1] - ((n + 1)*x + (9*n^2 + 30*n + 18)*x^2 + (9*n^3 + 153*n^2 + 288*n + 108)*x^3 + (108*n^3 + 540*n^2 + 864*n + 432)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (9*n^3 - 6*n)*x + (9*n^4 + 9*n^3)*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-e189038e95d51606","steps":["(\u03b8+1)\u00b2"],"edges":["e-4e4cfdea6d7e0f6f"],"text":"A304236 \u2192 (\u03b8+1)\u00b2"},"headline":"A304236 \u2192 (\u03b8+1)\u00b2","chain":null}