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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":"6f3a4b1925f0318665ed67416fa40c3b00b13ad2a8691eeea2ee0ab71f92f4c1","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-3f0a1785078e99be","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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:       5,        20;\n  2:      25,       280,         225;\n  3:     125,      2860,        6435,         2000;\n  4:     625,     25520,      115830,       102080,         15625;\n  5:    3125,    211140,     1674090,      2976160,       1319625,        112500;\n  6:   15625,   1665480,    21242655,     65475520,      59007375,      14989320,       765625;\n  7:   78125,  12720860,   247393845,   1211172080,    1892456375,     989575380,    155830535,      5000000;\n  8:  390625,  94983520,  2713117500,  19920062080,   49203865750,   44820139680,  14771417500,   1519736320,    31640625;\n  9: 1953125, 697697140, 28478602980, 301117122880, 1106526709750, 1593398462040, 922171188820, 202514510080, 14128367085, 195312500;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+12*n+9)*P(n,x) = -(60*n^2+240*n+225+(160*n^3+720*n^2+1016*n+348)*x-(144*n^4+832*n^3+1688*n^2+1392*n+369)*x^2)*P(n-2,x) - ((80*n^2+320*n+384)*x-(160*n^3+944*n^2+1832*n+1284)*x^2-(64*n^4+160*n^3-352*n^2-1272*n-900)*x^3)*d/dx P(n-2,x) - ((16*n^2+64*n+84)*x^2-(64*n^3+352*n^2+640*n+408)*x^3+(64*n^3+336*n^2+576*n+324)*x^4)*d^2/dx^2 P(n-2,x) + (32*n^2+108*n+90+(32*n^3+144*n^2+216*n+108)*x)*P(n-1,x) + ((16*n^2+56*n+60)*x-(32*n^3+176*n^2+312*n+180)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 20*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 20*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(60*n^2 + 240*n + 225 + (160*n^3 + 720*n^2 + 1016*n + 348)*x - (144*n^4 + 832*n^3 + 1688*n^2 + 1392*n + 369)*x^2)*P[n - 2, x] - ((80*n^2 + 320*n + 384)*x - (160*n^3 + 944*n^2 + 1832*n + 1284)*x^2 - (64*n^4 + 160*n^3 - 352*n^2 - 1272*n - 900)*x^3)*D[P[n - 2, x], x] - ((16*n^2 + 64*n + 84)*x^2 - (64*n^3 + 352*n^2 + 640*n + 408)*x^3 + (64*n^3 + 336*n^2 + 576*n + 324)*x^4)*D[P[n - 2, x], {x, 2}] + (32*n^2 + 108*n + 90 + (32*n^3 + 144*n^2 + 216*n + 108)*x)*P[n - 1, x] + ((16*n^2 + 56*n + 60)*x - (32*n^3 + 176*n^2 + 312*n + 180)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 12*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 20*x; for(n = 2, N, my(i = n + 1); v[i] = (-(60*n^2 + 240*n + 225 + (160*n^3 + 720*n^2 + 1016*n + 348)*x - (144*n^4 + 832*n^3 + 1688*n^2 + 1392*n + 369)*x^2)*v[i - 2] - ((80*n^2 + 320*n + 384)*x - (160*n^3 + 944*n^2 + 1832*n + 1284)*x^2 - (64*n^4 + 160*n^3 - 352*n^2 - 1272*n - 900)*x^3)*deriv(v[i - 2], x) - ((16*n^2 + 64*n + 84)*x^2 - (64*n^3 + 352*n^2 + 640*n + 408)*x^3 + (64*n^3 + 336*n^2 + 576*n + 324)*x^4)*deriv(deriv(v[i - 2], x), x) + (32*n^2 + 108*n + 90 + (32*n^3 + 144*n^2 + 216*n + 108)*x)*v[i - 1] + ((16*n^2 + 56*n + 60)*x - (32*n^3 + 176*n^2 + 312*n + 180)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 12*n + 9)); 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-3f0a1785078e99be","steps":["(\u03b8+1)\u00b2"],"edges":["e-e1eedbba3004a011"],"text":"A257615 \u2192 (\u03b8+1)\u00b2"},"headline":"A257615 \u2192 (\u03b8+1)\u00b2","chain":null}