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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":"1c4dc047d227e6624b0fa5277796d10b5eb1b252090e90dc8e66ed858164366b","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-39736a4ef132a31d","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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:       6,         8;\n  2:      35,        96,        36;\n  3:     204,       848,       648,       128;\n  4:    1189,      6624,      7668,      3072,       400;\n  5:    6930,     48408,     75168,     45568,     12000,      1152;\n  6:   40391,    339264,    661176,    537600,    214000,     41472,     3136;\n  7:  235416,   2310304,   5418576,   5530880,   2952000,    864000,   131712,     8192;\n  8: 1372105,  15405888,  42246648,  51913728,  34782000,  13644288,  3139136,   393216,   20736;\n  9: 7997214, 101101864, 317387808, 456161792, 367970400, 181173888, 55845888, 10551296, 1119744, 51200;\n","recurrence_sentence":"The row polynomials satisfy (256*n+24*n*x-(80*n^3-9*n)*x^2-15*n^3*x^3+n^5*x^4)*P(n,x) = (768*n+768+(448*n^2+1096*n+648)*x-(240*n^3+264*n^2-123*n-147)*x^2-(16*n^4+181*n^3+301*n^2+148*n+12)*x^3+(3*n^5+3*n^4-27*n^3-54*n^2-33*n-6)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (512+576*x-(448*n^2+448*n-152)*x^2-(456*n^2+456*n+42)*x^3+(8*n^4+16*n^3-138*n^2-146*n-25)*x^4+(6*n^4+12*n^3-9*n^2-15*n-3)*x^5+(n^4+2*n^3+n^2)*x^6)*d/dx 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[((768*n + 768 + (448*n^2 + 1096*n + 648)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (16*n^4 + 181*n^3 + 301*n^2 + 148*n + 12)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (512 + 576*x - (448*n^2 + 448*n - 152)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (8*n^4 + 16*n^3 - 138*n^2 - 146*n - 25)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(256*n + 24*n*x - (80*n^3 - 9*n)*x^2 - 15*n^3*x^3 + n^5*x^4)]];\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] = ((768*n + 768 + (448*n^2 + 1096*n + 648)*x - (240*n^3 + 264*n^2 - 123*n - 147)*x^2 - (16*n^4 + 181*n^3 + 301*n^2 + 148*n + 12)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (512 + 576*x - (448*n^2 + 448*n - 152)*x^2 - (456*n^2 + 456*n + 42)*x^3 + (8*n^4 + 16*n^3 - 138*n^2 - 146*n - 25)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(256*n + 24*n*x - (80*n^3 - 9*n)*x^2 - 15*n^3*x^3 + n^5*x^4)); 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-c2b7c74f8491f01b","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-63a398c0ccd31fef","e-d3c2a32ccedfaa69"],"text":"A207824 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A207824 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}