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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":"d4dddb915241517af6e467a9fde18504b0dc36064b18e0d7149ced9cd0beabe7","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-04fe76d5a46c2282","role":"sequence","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-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:      4;\n  2:     16,       2;\n  3:     64,      24;\n  4:    256,     192,      3;\n  5:   1024,    1280,     60;\n  6:   4096,    7680,    720,     4;\n  7:  16384,   43008,   6720,   112;\n  8:  65536,  229376,  53760,  1792,   5;\n  9: 262144, 1179648, 387072, 21504, 180;\n","recurrence_sentence":"The row polynomials satisfy -(384*n-384-(4*n^3-4*n)*x)*P(n,x) = -(1536*n-1536-(16*n^3-208*n+192)*x-(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (3072*x-(32*n^2+96*n+256)*x^2+(2*n^2+6*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(1536*n - 1536 - (16*n^3 - 208*n + 192)*x - (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (3072*x - (32*n^2 + 96*n + 256)*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(-384*n + 384 + (4*n^3 - 4*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(1536*n - 1536 - (16*n^3 - 208*n + 192)*x - (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (3072*x - (32*n^2 + 96*n + 256)*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(-384*n + 384 + (4*n^3 - 4*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-3dcaebc50e901d48","steps":["Veronese 2","\u03b8+1"],"edges":["e-d1b315cae6550577","e-f546b78c2a75f4e9"],"text":"A038231 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A038231 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}