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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":"7a5e9581661408caf814dcffa4d37caa6c208c7a454e51d68656ae65f5d28718","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-916d0051af2e0118","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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,  16;\n  2: 1,  48,   108;\n  3: 1,  96,   648,     512;\n  4: 1, 160,  2160,    5120,     2000;\n  5: 1, 240,  5400,   25600,    30000,     6912;\n  6: 1, 336, 11340,   89600,   210000,   145152,    21952;\n  7: 1, 448, 21168,  250880,   980000,  1354752,   614656,    65536;\n  8: 1, 576, 36288,  602112,  3528000,  8128512,  7375872,  2359296,  186624;\n  9: 1, 720, 58320, 1290240, 10584000, 36578304, 55319040, 35389440, 8398080, 512000;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-(8*n^4+12*n^3+2*n^2-2*n)*x+(4*n^6+12*n^5+12*n^4+4*n^3)*x^2)*P(n,x) = (n^2+n-(8*n^4+12*n^3-22*n^2-10*n)*x+(4*n^6+12*n^5-72*n^4-172*n^3-48*n^2-12*n)*x^2+(24*n^6+120*n^5+192*n^4+120*n^3+48*n^2+8*n)*x^3)*P(n-1,x) + ((2*n+1)*x-(16*n^3+40*n^2+24*n+6)*x^2+(8*n^5+44*n^4+112*n^3+140*n^2+64*n+12)*x^3-(16*n^5+88*n^4+160*n^3+120*n^2+48*n+8)*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 - (8*n^4 + 12*n^3 - 22*n^2 - 10*n)*x + (4*n^6 + 12*n^5 - 72*n^4 - 172*n^3 - 48*n^2 - 12*n)*x^2 + (24*n^6 + 120*n^5 + 192*n^4 + 120*n^3 + 48*n^2 + 8*n)*x^3)*P[n - 1, x] + ((2*n + 1)*x - (16*n^3 + 40*n^2 + 24*n + 6)*x^2 + (8*n^5 + 44*n^4 + 112*n^3 + 140*n^2 + 64*n + 12)*x^3 - (16*n^5 + 88*n^4 + 160*n^3 + 120*n^2 + 48*n + 8)*x^4)*D[P[n - 1, x], x])/(n^2 + n - (8*n^4 + 12*n^3 + 2*n^2 - 2*n)*x + (4*n^6 + 12*n^5 + 12*n^4 + 4*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 - (8*n^4 + 12*n^3 - 22*n^2 - 10*n)*x + (4*n^6 + 12*n^5 - 72*n^4 - 172*n^3 - 48*n^2 - 12*n)*x^2 + (24*n^6 + 120*n^5 + 192*n^4 + 120*n^3 + 48*n^2 + 8*n)*x^3)*v[i - 1] + ((2*n + 1)*x - (16*n^3 + 40*n^2 + 24*n + 6)*x^2 + (8*n^5 + 44*n^4 + 112*n^3 + 140*n^2 + 64*n + 12)*x^3 - (16*n^5 + 88*n^4 + 160*n^3 + 120*n^2 + 48*n + 8)*x^4)*deriv(v[i - 1], x))/(n^2 + n - (8*n^4 + 12*n^3 + 2*n^2 - 2*n)*x + (4*n^6 + 12*n^5 + 12*n^4 + 4*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-8ce61d302cec295b","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-2a32d110f0351649","e-7a02bcd7581ecac9"],"text":"A114687 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A114687 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}