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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":"d62031931cab00ea34c44bf96bcce987ef2160c17caff1ddaa2bebebf934e92e","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-7b4f74988046113d","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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,  40,   135;\n  3: 1,  72,   567,    896;\n  4: 1, 112,  1512,   5376,    5250;\n  5: 1, 160,  3240,  19200,   41250,    28512;\n  6: 1, 216,  6075,  52800,  185625,   277992,   147147;\n  7: 1, 280, 10395, 123200,  625625,  1513512,  1716715,   732160;\n  8: 1, 352, 16632, 256256, 1751750,  6054048, 10986976,  9957376,  3544398;\n  9: 1, 432, 25272, 489216, 4299750, 19813248, 50939616, 73322496, 55099278, 16796000;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n-(4*n^4+12*n^3+7*n^2-2*n)*x+(n^6+4*n^5+5*n^4+2*n^3)*x^2)*P(n,x) = (n^2+2*n-(4*n^4+12*n^3-n^2-26*n-16)*x+(n^6+4*n^5-9*n^4-68*n^3-126*n^2-98*n-28)*x^2+(2*n^6+14*n^5+40*n^4+60*n^3+50*n^2+22*n+4)*x^3)*P(n-1,x) + ((2*n+2)*x-(8*n^3+24*n^2+22*n+6)*x^2+(2*n^5+10*n^4+12*n^3-4*n^2-14*n-6)*x^3+(2*n^5+10*n^4+20*n^3+20*n^2+10*n+2)*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 + 2*n - (4*n^4 + 12*n^3 - n^2 - 26*n - 16)*x + (n^6 + 4*n^5 - 9*n^4 - 68*n^3 - 126*n^2 - 98*n - 28)*x^2 + (2*n^6 + 14*n^5 + 40*n^4 + 60*n^3 + 50*n^2 + 22*n + 4)*x^3)*P[n - 1, x] + ((2*n + 2)*x - (8*n^3 + 24*n^2 + 22*n + 6)*x^2 + (2*n^5 + 10*n^4 + 12*n^3 - 4*n^2 - 14*n - 6)*x^3 + (2*n^5 + 10*n^4 + 20*n^3 + 20*n^2 + 10*n + 2)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n - (4*n^4 + 12*n^3 + 7*n^2 - 2*n)*x + (n^6 + 4*n^5 + 5*n^4 + 2*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 + 2*n - (4*n^4 + 12*n^3 - n^2 - 26*n - 16)*x + (n^6 + 4*n^5 - 9*n^4 - 68*n^3 - 126*n^2 - 98*n - 28)*x^2 + (2*n^6 + 14*n^5 + 40*n^4 + 60*n^3 + 50*n^2 + 22*n + 4)*x^3)*v[i - 1] + ((2*n + 2)*x - (8*n^3 + 24*n^2 + 22*n + 6)*x^2 + (2*n^5 + 10*n^4 + 12*n^3 - 4*n^2 - 14*n - 6)*x^3 + (2*n^5 + 10*n^4 + 20*n^3 + 20*n^2 + 10*n + 2)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n - (4*n^4 + 12*n^3 + 7*n^2 - 2*n)*x + (n^6 + 4*n^5 + 5*n^4 + 2*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-66e2c3448245b0b7","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-10c4942a23949b48","e-9ed9d0f945bbd42f"],"text":"A033282 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A033282 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}