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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":"cea962091896c8bd824688556e612dadf5ebaee14b00cd3e67cd18c70a815213","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-76276d62adb9bc90","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","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:           20,            36;\n  2:          400,          1440,            972;\n  3:         8000,         43200,          58320,           23328;\n  4:       160000,       1152000,        2332800,         1866240,          524880;\n  5:      3200000,      28800000,       77760000,        93312000,        52488000,        11337408;\n  6:     64000000,     691200000,     2332800000,      3732480000,      3149280000,      1360488960,       238085568;\n  7:   1280000000,   16128000000,    65318400000,    130636800000,    146966400000,     95234227200,     33331979520,     4897760256;\n  8:  25600000000,  368640000000,  1741824000000,   4180377600000,   5878656000000,   5079158784000,   2666558361600,   783641640960,    99179645184;\n  9: 512000000000, 8294400000000, 44789760000000, 125411328000000, 211631616000000, 228562145280000, 159993501696000, 70527747686400, 17852336133120, 1983592903680;\n","recurrence_sentence":"The row polynomials satisfy (10+9*n*x)*P(n,x) = (200+(180*n+360)*x+(162*n+162)*x^2)*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[((200 + (180*n + 360)*x + (162*n + 162)*x^2)*P[n - 1, x])/(10 + 9*n*x)]];\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] = ((200 + (180*n + 360)*x + (162*n + 162)*x^2)*v[i - 1])/(10 + 9*n*x)); 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-4e096b5929fd3f2e","steps":["basis A013609","\u03b8+1"],"edges":["e-66940ce826511afb","e-b4f6bc866899c6ec"],"text":"A038323 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038323 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}