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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":"42b3acebf8946e83246822cf69558fdd0e1d5a763abb62ca0406ef031d4f1ea9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5c56442adc0376c2","role":"sequence","name":"Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:            1;\n   2:            5,            4;\n   3:           45,           60,            9;\n   4:          585,         1020,          270,           16;\n   5:         9945,        20700,         7425,          800,          25;\n   6:       208845,       495180,       217350,        32400,        1875,         36;\n   7:      5221125,     13710060,      6982605,      1293600,      105000,       3780,       49;\n   8:    151412625,    432186300,    247308390,     53808720,     5486250,     279720,     6860,      64;\n   9:   4996616625,  15299984700,   9628212825,   2376773280,   287067375,   18529560,   648270,   11520,    81;\n  10: 184874815125, 601385885100, 409925265750, 112187754000, 15483470625, 1191618540, 53096400, 1353600, 18225, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-60*n+56+(16*n-28)*x-9*x^2)*P(n-2,x) - ((8*n-14)*x-(16*n-28)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (8*n-11+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 60*n + 56 + (16*n - 28)*x - 9*x^2)*P[n - 2, x] - ((8*n - 14)*x - (16*n - 28)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (8*n - 11 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 4*x; for(n = 3, N, my(i = n); v[i] = -(16*n^2 - 60*n + 56 + (16*n - 28)*x - 9*x^2)*v[i - 2] - ((8*n - 14)*x - (16*n - 28)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (8*n - 11 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5c56442adc0376c2","steps":["(\u03b8+1)\u00b2"],"edges":["e-6cbc1b290d21bbfb"],"text":"A049029 \u2192 (\u03b8+1)\u00b2"},"headline":"A049029 \u2192 (\u03b8+1)\u00b2","chain":null}