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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":"43a5d5b681c88ed6ed23d015bf2cc99df3af46b017b9435c41a7fe2078e036bc","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-e8e64448d4c6498a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 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: 0, 8;\n  2: 0, 4, 27;\n  3: 0, 0, 36,  64;\n  4: 0, 0,  9, 160, 125;\n  5: 0, 0,  0,  96, 500,  216;\n  6: 0, 0,  0,  16, 525, 1260,  343;\n  7: 0, 0,  0,   0, 200, 2016, 2744,   512;\n  8: 0, 0,  0,   0,  25, 1296, 6174,  5376,  729;\n  9: 0, 0,  0,   0,   0,  360, 5880, 16128, 9720, 1000;\n","recurrence_sentence":"The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n-(8*n^5+36*n^4+34*n^3-2*n^2-4*n)*x+(16*n^5+16*n^4+4*n^3)*x^2)*P(n,x) = -(n^5+5*n^4+7*n^3-n^2-8*n-4-(11*n^5+65*n^4+127*n^3+91*n^2+6*n-12)*x+(40*n^5+212*n^4+342*n^3+104*n^2-110*n-12)*x^2-(48*n^5+160*n^4+124*n^3-20*n^2-20*n-4)*x^3)*P(n-1,x) + ((2*n^4+12*n^3+26*n^2+24*n+8)*x-(18*n^4+108*n^3+214*n^2+156*n+24)*x^2+(48*n^4+224*n^3+340*n^2+188*n+24)*x^3-(32*n^4+128*n^3+152*n^2+56*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^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (11*n^5 + 65*n^4 + 127*n^3 + 91*n^2 + 6*n - 12)*x + (40*n^5 + 212*n^4 + 342*n^3 + 104*n^2 - 110*n - 12)*x^2 - (48*n^5 + 160*n^4 + 124*n^3 - 20*n^2 - 20*n - 4)*x^3)*P[n - 1, x] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (18*n^4 + 108*n^3 + 214*n^2 + 156*n + 24)*x^2 + (48*n^4 + 224*n^3 + 340*n^2 + 188*n + 24)*x^3 - (32*n^4 + 128*n^3 + 152*n^2 + 56*n + 8)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (8*n^5 + 36*n^4 + 34*n^3 - 2*n^2 - 4*n)*x + (16*n^5 + 16*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^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4 - (11*n^5 + 65*n^4 + 127*n^3 + 91*n^2 + 6*n - 12)*x + (40*n^5 + 212*n^4 + 342*n^3 + 104*n^2 - 110*n - 12)*x^2 - (48*n^5 + 160*n^4 + 124*n^3 - 20*n^2 - 20*n - 4)*x^3)*v[i - 1] + ((2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x - (18*n^4 + 108*n^3 + 214*n^2 + 156*n + 24)*x^2 + (48*n^4 + 224*n^3 + 340*n^2 + 188*n + 24)*x^3 - (32*n^4 + 128*n^3 + 152*n^2 + 56*n + 8)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (8*n^5 + 36*n^4 + 34*n^3 - 2*n^2 - 4*n)*x + (16*n^5 + 16*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-e8e64448d4c6498a","steps":["(\u03b8+1)\u00b2"],"edges":["e-38856a9af7e27c4a"],"text":"A119900 \u2192 (\u03b8+1)\u00b2"},"headline":"A119900 \u2192 (\u03b8+1)\u00b2","chain":null}