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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":"fcfef07a99e0aac6883235db8e389c9fe285a95ec161839f412fd9be765bd4b1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-aff30617856e629d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).","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:     4,      4;\n  2:    14,     28,      9;\n  3:    44,    132,     90,     16;\n  4:   128,    512,    540,    208,     25;\n  5:   352,   1760,   2520,   1520,    400,     36;\n  6:   928,   5568,  10080,   8320,   3450,    684,    49;\n  7:  2368,  16576,  36288,  38080,  21700,   6804,  1078,    64;\n  8:  5888,  47104, 120960, 154112, 112000,  48384, 12152,  1600,   81;\n  9: 14336, 129024, 380160, 569856, 504000, 278208, 96432, 20160, 2268, 100;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+2*n+4+(n^3+3*n^2+4*n+4)*x+(n^3+7*n-2)*x^2+n^2*x^3)*P(n,x) = (4*n^2+12*n+16+(2*n^3+14*n^2+32*n+32)*x+(3*n^3+12*n^2+33*n+24)*x^2+(n^3+5*n^2+14*n+8)*x^3+(n^2+2*n+1)*x^4)*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[((4*n^2 + 12*n + 16 + (2*n^3 + 14*n^2 + 32*n + 32)*x + (3*n^3 + 12*n^2 + 33*n + 24)*x^2 + (n^3 + 5*n^2 + 14*n + 8)*x^3 + (n^2 + 2*n + 1)*x^4)*P[n - 1, x])/(2*n^2 + 2*n + 4 + (n^3 + 3*n^2 + 4*n + 4)*x + (n^3 + 7*n - 2)*x^2 + n^2*x^3)]];\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] = ((4*n^2 + 12*n + 16 + (2*n^3 + 14*n^2 + 32*n + 32)*x + (3*n^3 + 12*n^2 + 33*n + 24)*x^2 + (n^3 + 5*n^2 + 14*n + 8)*x^3 + (n^2 + 2*n + 1)*x^4)*v[i - 1])/(2*n^2 + 2*n + 4 + (n^3 + 3*n^2 + 4*n + 4)*x + (n^3 + 7*n - 2)*x^2 + n^2*x^3)); 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-c8c19eba9d68881f","steps":["\u2212D\u00b2+(\u22121+n)D+1+n","x \u21a6 x+1","\u03b8+1"],"edges":["e-168270346f6f7c93","e-2c04e0d67d228006","e-0890eba5db89172d"],"text":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}