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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":"b2c0b130fd29c45211fa1fc55808a5f2cde3c40c6a5d4a7feff1e07cd95ef136","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-4ca874876022e136","role":"sequence","name":"Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","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,  12;\n  2: 1,  32,    54;\n  3: 1,  60,   270,    160;\n  4: 1,  96,   810,   1280,     375;\n  5: 1, 140,  1890,   5600,    4375,     756;\n  6: 1, 192,  3780,  17920,   26250,   12096,    1372;\n  7: 1, 252,  6804,  47040,  110250,   95256,   28812,   2304;\n  8: 1, 320, 11340, 107520,  367500,  508032,  288120,  61440,   3645;\n  9: 1, 396, 17820, 221760, 1039500, 2095632, 1901592, 760320, 120285, 5500;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n-(4*n^4+14*n^3+10*n^2-4*n)*x+(n^6+6*n^5+12*n^4+8*n^3)*x^2)*P(n,x) = (n^2+2*n-(4*n^4+14*n^3-2*n^2-20*n-8)*x+(n^6+6*n^5-9*n^4-78*n^3-99*n^2-66*n-16)*x^2+(3*n^6+24*n^5+69*n^4+96*n^3+84*n^2+40*n+8)*x^3)*P(n-1,x) + ((2*n+2)*x-(8*n^3+32*n^2+32*n+12)*x^2+(2*n^5+16*n^4+52*n^3+84*n^2+62*n+18)*x^3-(2*n^5+16*n^4+44*n^3+52*n^2+32*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^2 + 2*n - (4*n^4 + 14*n^3 - 2*n^2 - 20*n - 8)*x + (n^6 + 6*n^5 - 9*n^4 - 78*n^3 - 99*n^2 - 66*n - 16)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*P[n - 1, x] + ((2*n + 2)*x - (8*n^3 + 32*n^2 + 32*n + 12)*x^2 + (2*n^5 + 16*n^4 + 52*n^3 + 84*n^2 + 62*n + 18)*x^3 - (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n - (4*n^4 + 14*n^3 + 10*n^2 - 4*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*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 + 14*n^3 - 2*n^2 - 20*n - 8)*x + (n^6 + 6*n^5 - 9*n^4 - 78*n^3 - 99*n^2 - 66*n - 16)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*v[i - 1] + ((2*n + 2)*x - (8*n^3 + 32*n^2 + 32*n + 12)*x^2 + (2*n^5 + 16*n^4 + 52*n^3 + 84*n^2 + 62*n + 18)*x^3 - (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n - (4*n^4 + 14*n^3 + 10*n^2 - 4*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*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-4ca874876022e136","steps":["(\u03b8+1)\u00b2"],"edges":["e-a3706a3507b0ac5d"],"text":"A062196 \u2192 (\u03b8+1)\u00b2"},"headline":"A062196 \u2192 (\u03b8+1)\u00b2","chain":null}