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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":"dedf49350636d1eb16874d1a24856d4f93be18386540d708718a22178be81819","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-ca41bee2bc8fcdbe","role":"sequence","name":"The convolution matrix of the double factorial of odd numbers (A001147).","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:         3,          4;\n  2:        15,         36,          9;\n  3:       105,        348,        162,         16;\n  4:       945,       3900,       2565,        480,        25;\n  5:     10395,      50580,      42120,      11280,      1125,        36;\n  6:    135135,     749700,     745605,     255360,     36750,      2268,       49;\n  7:   2027025,   12535740,   14362110,    5922000,   1097250,     98280,     4116,     64;\n  8:  34459425,  233751420,  301085505,  143972640,  32295375,   3742200,   228438,   6912,    81;\n  9: 654729075, 4812866100, 6848736300, 3702661200, 967750875, 136318140, 10804500, 478080, 10935, 100;\n","recurrence_sentence":"The row polynomials satisfy (8*n-6*n*x+4*n^2*x^2)*P(n,x) = (16*n^2+8*n-(12*n^2-14*n-12)*x+(8*n^3+4*n^2-12*n-12)*x^2+(4*n^2+8*n+4)*x^3)*P(n-1,x) + ((8*n+9)*x-(12*n+13)*x^2+(4*n^2+12*n+6)*x^3)*d/dx P(n-1,x) + (2*x-3*x^2+(4*n+2)*x^3)*d^2/dx^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[((16*n^2 + 8*n - (12*n^2 - 14*n - 12)*x + (8*n^3 + 4*n^2 - 12*n - 12)*x^2 + (4*n^2 + 8*n + 4)*x^3)*P[n - 1, x] + ((8*n + 9)*x - (12*n + 13)*x^2 + (4*n^2 + 12*n + 6)*x^3)*D[P[n - 1, x], x] + (2*x - 3*x^2 + (4*n + 2)*x^3)*D[P[n - 1, x], {x, 2}])/(8*n - 6*n*x + 4*n^2*x^2)]];\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] = ((16*n^2 + 8*n - (12*n^2 - 14*n - 12)*x + (8*n^3 + 4*n^2 - 12*n - 12)*x^2 + (4*n^2 + 8*n + 4)*x^3)*v[i - 1] + ((8*n + 9)*x - (12*n + 13)*x^2 + (4*n^2 + 12*n + 6)*x^3)*deriv(v[i - 1], x) + (2*x - 3*x^2 + (4*n + 2)*x^3)*deriv(deriv(v[i - 1], x), x))/(8*n - 6*n*x + 4*n^2*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-ca41bee2bc8fcdbe","steps":["(\u03b8+1)\u00b2"],"edges":["e-54ed4ebabe501e59"],"text":"A035342 \u2192 (\u03b8+1)\u00b2"},"headline":"A035342 \u2192 (\u03b8+1)\u00b2","chain":null}