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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":"5e6f113fe643420e0e0fa7a3a005eb393a846cee77e21cf28736635eaf13caaf","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-c914b69838ac70ec","role":"sequence","name":"Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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,   8;\n  3: 1,  24;\n  4: 1,  48,   108;\n  5: 1,  80,   540;\n  6: 1, 120,  1620,   1920;\n  7: 1, 168,  3780,  13440;\n  8: 1, 224,  7560,  53760,  42000;\n  9: 1, 288, 13608, 161280, 378000;\n","recurrence_sentence":"The row polynomials satisfy (1+(3*n^2-n-2)*x+(n^4+2*n^3+n^2)*x^2)*P(n,x) = (1+(3*n^2+7*n-10)*x+(n^4+12*n^3+5*n^2-18*n+4)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) - (4*x^2+(12*n^2+20*n)*x^3+(4*n^4+24*n^3+52*n^2+48*n+16)*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[((1 + (3*n^2 + 7*n - 10)*x + (n^4 + 12*n^3 + 5*n^2 - 18*n + 4)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] - (4*x^2 + (12*n^2 + 20*n)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(1 + (3*n^2 - n - 2)*x + (n^4 + 2*n^3 + n^2)*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] = ((1 + (3*n^2 + 7*n - 10)*x + (n^4 + 12*n^3 + 5*n^2 - 18*n + 4)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] - (4*x^2 + (12*n^2 + 20*n)*x^3 + (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(1 + (3*n^2 - n - 2)*x + (n^4 + 2*n^3 + n^2)*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-c914b69838ac70ec","steps":["(\u03b8+1)\u00b2"],"edges":["e-5ad1e40b769264f9"],"text":"A059344 \u2192 (\u03b8+1)\u00b2"},"headline":"A059344 \u2192 (\u03b8+1)\u00b2","chain":null}