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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":"fb781de301b888f635a71324bbffcbe8d28de121b70edfcf560109e1ed254b6d","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-2245b35ec7cd4857","role":"sequence","name":"One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:  2,    8;\n  2:  3,   40,    27;\n  3:  4,  112,   252,     64;\n  4:  5,  240,  1134,    960,     125;\n  5:  6,  440,  3564,   6336,    2750,     216;\n  6:  7,  728,  9009,  27456,   25025,    6552,     343;\n  7:  8, 1120, 19656,  91520,  143000,   78624,   13720,    512;\n  8:  9, 1632, 38556, 254592,  607750,  572832,  209916,  26112,   729;\n  9: 10, 2280, 69768, 620160, 2099500, 3023280, 1899240, 496128, 46170, 1000;\n","recurrence_sentence":"The row polynomials satisfy (n-(10*n^3+n^2-n)*x+(4*n^5+4*n^4+n^3)*x^2)*P(n,x) = (n+1-(10*n^3+11*n^2-12*n+3)*x+(4*n^5+8*n^4-49*n^3-56*n^2+28*n+3)*x^2+(12*n^5+40*n^4+31*n^3-5*n^2-5*n-1)*x^3)*P(n-1,x) + (2*x-(20*n^2+32*n+6)*x^2+(8*n^4+32*n^3+58*n^2+46*n+6)*x^3-(8*n^4+32*n^3+38*n^2+14*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 + 1 - (10*n^3 + 11*n^2 - 12*n + 3)*x + (4*n^5 + 8*n^4 - 49*n^3 - 56*n^2 + 28*n + 3)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*P[n - 1, x] + (2*x - (20*n^2 + 32*n + 6)*x^2 + (8*n^4 + 32*n^3 + 58*n^2 + 46*n + 6)*x^3 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*D[P[n - 1, x], x])/(n - (10*n^3 + n^2 - n)*x + (4*n^5 + 4*n^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 + 1 - (10*n^3 + 11*n^2 - 12*n + 3)*x + (4*n^5 + 8*n^4 - 49*n^3 - 56*n^2 + 28*n + 3)*x^2 + (12*n^5 + 40*n^4 + 31*n^3 - 5*n^2 - 5*n - 1)*x^3)*v[i - 1] + (2*x - (20*n^2 + 32*n + 6)*x^2 + (8*n^4 + 32*n^3 + 58*n^2 + 46*n + 6)*x^3 - (8*n^4 + 32*n^3 + 38*n^2 + 14*n + 2)*x^4)*deriv(v[i - 1], x))/(n - (10*n^3 + n^2 - n)*x + (4*n^5 + 4*n^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-2245b35ec7cd4857","steps":["(\u03b8+1)\u00b2"],"edges":["e-988bd1603bc5c7fc"],"text":"A091044 \u2192 (\u03b8+1)\u00b2"},"headline":"A091044 \u2192 (\u03b8+1)\u00b2","chain":null}