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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":"c100c9e20f6679ea64c35ef37c4ee39ac7c00f2b3ffc9e83a42744ca136918f4","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-32779d2065847504","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.","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:  3,   16;\n  2:  6,   64,    90;\n  3: 10,  160,   450,    320;\n  4: 15,  320,  1350,   1920,    875;\n  5: 21,  560,  3150,   6720,   6125,   2016;\n  6: 28,  896,  6300,  17920,  24500,  16128,   4116;\n  7: 36, 1344, 11340,  40320,  73500,  72576,  37044,   7680;\n  8: 45, 1920, 18900,  80640, 183750, 241920, 185220,  76800,  13365;\n  9: 55, 2640, 29700, 147840, 404250, 665280, 679140, 422400, 147015, 22000;\n","recurrence_sentence":"The row polynomials satisfy (3*n+(13*n^2-4*n)*x+(8*n^3+2*n^2-n)*x^2+(n^4+2*n^3)*x^3)*P(n,x) = (3*n+6+(13*n^2+38*n+24)*x+(8*n^3+47*n^2+80*n+36)*x^2+(n^4+15*n^3+51*n^2+62*n+24)*x^3+(n^4+7*n^3+17*n^2+17*n+6)*x^4)*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[((3*n + 6 + (13*n^2 + 38*n + 24)*x + (8*n^3 + 47*n^2 + 80*n + 36)*x^2 + (n^4 + 15*n^3 + 51*n^2 + 62*n + 24)*x^3 + (n^4 + 7*n^3 + 17*n^2 + 17*n + 6)*x^4)*P[n - 1, x])/(3*n + (13*n^2 - 4*n)*x + (8*n^3 + 2*n^2 - n)*x^2 + (n^4 + 2*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] = ((3*n + 6 + (13*n^2 + 38*n + 24)*x + (8*n^3 + 47*n^2 + 80*n + 36)*x^2 + (n^4 + 15*n^3 + 51*n^2 + 62*n + 24)*x^3 + (n^4 + 7*n^3 + 17*n^2 + 17*n + 6)*x^4)*v[i - 1])/(3*n + (13*n^2 - 4*n)*x + (8*n^3 + 2*n^2 - n)*x^2 + (n^4 + 2*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-c8c19eba9d68881f","steps":["(1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2","(\u03b8+1)\u00b2"],"edges":["e-6ba022abf95075ed","e-9df3e4f5d2ab24b6"],"text":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 (\u03b8+1)\u00b2","chain":null}