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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":"647c7259c1b4281a4b0ed10a5f233c986f1d3f309583389088005967e68989da","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-a2769223501d377d","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^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:        36,         960,          3600;\n  1:       216,        8640,         64800,         128000;\n  2:      1296,       69120,        777600,        3072000,         4000000;\n  3:      7776,      518400,       7776000,       46080000,       120000000,        115200000;\n  4:     46656,     3732480,      69984000,      552960000,      2160000000,       4147200000,        3136000000;\n  5:    279936,    26127360,     587865600,     5806080000,     30240000000,      87091200000,      131712000000,       81920000000;\n  6:   1679616,   179159040,    4702924800,    55738368000,    362880000000,    1393459200000,     3161088000000,     3932160000000,     2073600000000;\n  7:  10077696,  1209323520,   36279705600,   501645312000,   3919104000000,   18811699200000,    56899584000000,   106168320000000,   111974400000000,     51200000000000;\n  8:  60466176,  8062156800,  272097792000,  4299816960000,  39191040000000,  225740390400000,   853493760000000,  2123366400000000,  3359232000000000,   3072000000000000,  1239040000000000;\n  9: 362797056, 53210234880, 1995383808000, 35473489920000, 369515520000000, 2483144294400000, 11266117632000000, 35035545600000000, 73903104000000000, 101376000000000000, 81776640000000000, 29491200000000000;\n","recurrence_sentence":"The row polynomials satisfy (9+(90*n+150)*x+(100*n^2+400*n+400)*x^2)*P(n,x) = (54+(540*n+1620)*x+(600*n^2+5400*n+10200)*x^2+(2000*n^2+12000*n+18000)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 960*x + 3600*x^2.","mathematica":"P[0, x_] := 36 + 960*x + 3600*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((54 + (540*n + 1620)*x + (600*n^2 + 5400*n + 10200)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*P[n - 1, x])/(9 + (90*n + 150)*x + (100*n^2 + 400*n + 400)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 36 + 960*x + 3600*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((54 + (540*n + 1620)*x + (600*n^2 + 5400*n + 10200)*x^2 + (2000*n^2 + 12000*n + 18000)*x^3)*v[i - 1])/(9 + (90*n + 150)*x + (100*n^2 + 400*n + 400)*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-bbfc5a54ff13a785","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-db3287eb465d1cf9","e-a8b6b7aaa50e3b9a"],"text":"A013612 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A013612 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}