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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":"e7215c1245e94bb39b277d443f9dce889cfe528125d9c869504d58ff208b7bd1","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-0b0f0dd4d2c8eb05","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (m*n - m*k + 1)*T(n - 1, k - 1) + k*(m*k - (m - 1))*T(n - 1, k) where m = 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,        4;\n  2: 1,       36,          9;\n  3: 1,      236,        378,           16;\n  4: 1,     1444,       8325,         2464,           25;\n  5: 1,     8700,     147618,       142992,        12675,           36;\n  6: 1,    52244,    2390445,      5840800,      1687500,        57132,          49;\n  7: 1,   313516,   37149714,    201789520,    139821250,     15920712,      237748,        64;\n  8: 1,  1881156,  566416053,   6376589856,   9129621375,   2460165192,   129984358,    939264,      81;\n  9: 1, 11287004, 8559729810, 191635020304, 520430600025, 280690795692, 35268483824, 961589120, 3578661, 100;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = ((16*n^2-16*n-12)*x+(72*n^3-124*n^2+14*n+15)*x^2)*P(n-2,x) + ((12*n+6)*x+(56*n^2-32*n-18)*x^2+(32*n^3-160*n^2+32*n+20)*x^3)*d/dx P(n-2,x) + ((18*n+9)*x^2+(120*n^2-32*n-18)*x^3-(32*n^2-8*n-4)*x^4)*d^2/dx^2 P(n-2,x) + ((4*n+2)*x^3+(32*n^2-8*n-4)*x^4)*d^3/dx^3 P(n-2,x) + (2*n-1+(16*n^2-16*n+4)*x)*P(n-1,x) + ((8*n-6)*x-(16*n^2-4)*x^2)*d/dx P(n-1,x) + (4*n-2)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((16*n^2 - 16*n - 12)*x + (72*n^3 - 124*n^2 + 14*n + 15)*x^2)*P[n - 2, x] + ((12*n + 6)*x + (56*n^2 - 32*n - 18)*x^2 + (32*n^3 - 160*n^2 + 32*n + 20)*x^3)*D[P[n - 2, x], x] + ((18*n + 9)*x^2 + (120*n^2 - 32*n - 18)*x^3 - (32*n^2 - 8*n - 4)*x^4)*D[P[n - 2, x], {x, 2}] + ((4*n + 2)*x^3 + (32*n^2 - 8*n - 4)*x^4)*D[P[n - 2, x], {x, 3}] + (2*n - 1 + (16*n^2 - 16*n + 4)*x)*P[n - 1, x] + ((8*n - 6)*x - (16*n^2 - 4)*x^2)*D[P[n - 1, x], x] + (4*n - 2)*x^2*D[P[n - 1, x], {x, 2}])/(2*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((16*n^2 - 16*n - 12)*x + (72*n^3 - 124*n^2 + 14*n + 15)*x^2)*v[i - 2] + ((12*n + 6)*x + (56*n^2 - 32*n - 18)*x^2 + (32*n^3 - 160*n^2 + 32*n + 20)*x^3)*deriv(v[i - 2], x) + ((18*n + 9)*x^2 + (120*n^2 - 32*n - 18)*x^3 - (32*n^2 - 8*n - 4)*x^4)*deriv(deriv(v[i - 2], x), x) + ((4*n + 2)*x^3 + (32*n^2 - 8*n - 4)*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n - 1 + (16*n^2 - 16*n + 4)*x)*v[i - 1] + ((8*n - 6)*x - (16*n^2 - 4)*x^2)*deriv(v[i - 1], x) + (4*n - 2)*x^2*deriv(deriv(v[i - 1], x), x))/(2*n - 1)); 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-0b0f0dd4d2c8eb05","steps":["(\u03b8+1)\u00b2"],"edges":["e-76091a1e3d816c6f"],"text":"A166961 \u2192 (\u03b8+1)\u00b2"},"headline":"A166961 \u2192 (\u03b8+1)\u00b2","chain":null}