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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":"62f62595680d60757c88f92c1c2ad2f273217d9e89a2b99e78e347da27bdab2c","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-dcd2d11144e6ea21","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) = 9*x + 1.","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,         4;\n  2: 1,        80,            9;\n  3: 1,       876,         1971,             16;\n  4: 1,      8872,        74898,          35488,             25;\n  5: 1,     88868,      1981998,        3523552,         555425,             36;\n  6: 1,    888864,     45056223,      197318912,      125156175,        7999776,             49;\n  7: 1,   8888860,    948065661,     8488627760,    13263480875,     3792262644,      108888535,            64;\n  8: 1,  88888856,  19113243984,   315212280224,   981497584750,   709227630504,   104060995024,    1422221696,          81;\n  9: 1, 888888852, 375951630960, 10694794369152, 58971293276850, 84918662318664, 32752807755528, 2673433820160, 17999999253, 100;\n","recurrence_sentence":"The row polynomials satisfy (81*n^2-144*n+64)*P(n,x) = (648*n^2-504*n-64-(2916*n^3-7776*n^2-19332*n+44608)*x+(59049*n^4-223074*n^3+269001*n^2-115344*n+9792)*x^2)*P(n-2,x) - ((1458*n^2-1134*n-6624)*x-(2916*n^3+21384*n^2-12123*n-37328)*x^2-(26244*n^4-185166*n^3+284067*n^2-139824*n+13504)*x^3)*d/dx P(n-2,x) - ((6561*n^2-5103*n-6480)*x^2-(26244*n^3-43740*n^2+22113*n-9360)*x^3+(26244*n^3-50301*n^2+27216*n-2880)*x^4)*d^2/dx^2 P(n-2,x) - (567*n^2-360*n-128-(2916*n^3-7776*n^2+6912*n-2048)*x)*P(n-1,x) + ((1458*n^2-1863*n-144)*x-(2916*n^3-4860*n^2+1728*n+256)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((648*n^2 - 504*n - 64 - (2916*n^3 - 7776*n^2 - 19332*n + 44608)*x + (59049*n^4 - 223074*n^3 + 269001*n^2 - 115344*n + 9792)*x^2)*P[n - 2, x] - ((1458*n^2 - 1134*n - 6624)*x - (2916*n^3 + 21384*n^2 - 12123*n - 37328)*x^2 - (26244*n^4 - 185166*n^3 + 284067*n^2 - 139824*n + 13504)*x^3)*D[P[n - 2, x], x] - ((6561*n^2 - 5103*n - 6480)*x^2 - (26244*n^3 - 43740*n^2 + 22113*n - 9360)*x^3 + (26244*n^3 - 50301*n^2 + 27216*n - 2880)*x^4)*D[P[n - 2, x], {x, 2}] - (567*n^2 - 360*n - 128 - (2916*n^3 - 7776*n^2 + 6912*n - 2048)*x)*P[n - 1, x] + ((1458*n^2 - 1863*n - 144)*x - (2916*n^3 - 4860*n^2 + 1728*n + 256)*x^2)*D[P[n - 1, x], x])/(81*n^2 - 144*n + 64)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((648*n^2 - 504*n - 64 - (2916*n^3 - 7776*n^2 - 19332*n + 44608)*x + (59049*n^4 - 223074*n^3 + 269001*n^2 - 115344*n + 9792)*x^2)*v[i - 2] - ((1458*n^2 - 1134*n - 6624)*x - (2916*n^3 + 21384*n^2 - 12123*n - 37328)*x^2 - (26244*n^4 - 185166*n^3 + 284067*n^2 - 139824*n + 13504)*x^3)*deriv(v[i - 2], x) - ((6561*n^2 - 5103*n - 6480)*x^2 - (26244*n^3 - 43740*n^2 + 22113*n - 9360)*x^3 + (26244*n^3 - 50301*n^2 + 27216*n - 2880)*x^4)*deriv(deriv(v[i - 2], x), x) - (567*n^2 - 360*n - 128 - (2916*n^3 - 7776*n^2 + 6912*n - 2048)*x)*v[i - 1] + ((1458*n^2 - 1863*n - 144)*x - (2916*n^3 - 4860*n^2 + 1728*n + 256)*x^2)*deriv(v[i - 1], x))/(81*n^2 - 144*n + 64)); 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-dcd2d11144e6ea21","steps":["(\u03b8+1)\u00b2"],"edges":["e-81554e707d87df1d"],"text":"A257608 \u2192 (\u03b8+1)\u00b2"},"headline":"A257608 \u2192 (\u03b8+1)\u00b2","chain":null}