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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":"f7ee9cf8829859724557058bbba47f98861b59243e998dae74ce71f6b524097a","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-c85abc56078671cc","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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,   8;\n  2: 1,  32,    27;\n  3: 1,  72,   225,     64;\n  4: 1, 128,   882,    896,     125;\n  5: 1, 200,  2430,   5376,    2625,     216;\n  6: 1, 288,  5445,  21120,   22275,    6336,     343;\n  7: 1, 392, 10647,  64064,  117975,   72072,   13377,    512;\n  8: 1, 512, 18900, 163072,  464750,  494208,  196196,  25600,   729;\n  9: 1, 648, 31212, 365568, 1491750, 2450448, 1689324, 470016, 45441, 1000;\n","recurrence_sentence":"The row polynomials satisfy -(36*n-27-(88*n^3-114*n^2+29*n+24)*x+(16*n^5-12*n^4+14*n^3+16*n^2-4*n-3)*x^2-(4*n^5+4*n^4+n^3)*x^3)*P(n,x) = -(36*n-27-(88*n^3-114*n^2-403*n+384)*x+(16*n^5-12*n^4-442*n^3+76*n^2+535*n-362)*x^2+(44*n^5+84*n^4+29*n^3+49*n^2+45*n-44)*x^3-(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^4)*P(n-1,x) - (72*x-(176*n^2+168*n-34)*x^2+(32*n^4+96*n^3+252*n^2+196*n-78)*x^3-(40*n^4+128*n^3+114*n^2+42*n+30)*x^4+(8*n^4+32*n^3+38*n^2+14*n+2)*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[(-(36*n - 27 - (88*n^3 - 114*n^2 - 403*n + 384)*x + (16*n^5 - 12*n^4 - 442*n^3 + 76*n^2 + 535*n - 362)*x^2 + (44*n^5 + 84*n^4 + 29*n^3 + 49*n^2 + 45*n - 44)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*P[n - 1, x] - (72*x - (176*n^2 + 168*n - 34)*x^2 + (32*n^4 + 96*n^3 + 252*n^2 + 196*n - 78)*x^3 - (40*n^4 + 128*n^3 + 114*n^2 + 42*n + 30)*x^4 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*D[P[n - 1, x], x])/(-36*n + 27 + (88*n^3 - 114*n^2 + 29*n + 24)*x - (16*n^5 - 12*n^4 + 14*n^3 + 16*n^2 - 4*n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*x^3)]];\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] = (-(36*n - 27 - (88*n^3 - 114*n^2 - 403*n + 384)*x + (16*n^5 - 12*n^4 - 442*n^3 + 76*n^2 + 535*n - 362)*x^2 + (44*n^5 + 84*n^4 + 29*n^3 + 49*n^2 + 45*n - 44)*x^3 - (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^4)*v[i - 1] - (72*x - (176*n^2 + 168*n - 34)*x^2 + (32*n^4 + 96*n^3 + 252*n^2 + 196*n - 78)*x^3 - (40*n^4 + 128*n^3 + 114*n^2 + 42*n + 30)*x^4 + (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^5)*deriv(v[i - 1], x))/(-36*n + 27 + (88*n^3 - 114*n^2 + 29*n + 24)*x - (16*n^5 - 12*n^4 + 14*n^3 + 16*n^2 - 4*n - 3)*x^2 + (4*n^5 + 4*n^4 + n^3)*x^3)); 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-c85abc56078671cc","steps":["(\u03b8+1)\u00b2"],"edges":["e-ba353ec9512d926d"],"text":"A232535 \u2192 (\u03b8+1)\u00b2"},"headline":"A232535 \u2192 (\u03b8+1)\u00b2","chain":null}