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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":"245baf25335d6707d83237b037832a707cfc86f849b0d7878ec9d7919ab38bc1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e2ca539e0aa62c2f","role":"sequence","name":"basis_map_A013611 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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:  1,    8;\n   3:  2,   24,    48;\n   4:  3,   56,   240,    256;\n   5:  5,  120,   768,   1792,    1280;\n   6:  8,  240,  2064,   7424,   11520,    6144;\n   7: 13,  464,  4992,  24320,   58880,   67584,    28672;\n   8: 21,  872, 11280,  69376,  229120,  411648,   372736,   131072;\n   9: 34, 1608, 24288, 180480,  756480, 1861632,  2637824,  1966080,   589824;\n  10: 55, 2920, 50448, 439552, 2234880, 7004160, 13619200, 15859712, 10027008, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (40*n-40+(80*n^2+48*n-128)*x+(64*n^3+64*n^2-128*n)*x^2)*P(n,x) = -(80*n-200+(240*n^2-464*n-640)*x+(128*n^3-64*n^2-1600*n-384)*x^2-(512*n^2+1024*n)*x^3)*P(n-1,x) + (25+(100*n+180)*x+(80*n^2+640*n+368)*x^2+(384*n^2+1088*n+192)*x^3+(256*n^2+512*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(80*n - 200 + (240*n^2 - 464*n - 640)*x + (128*n^3 - 64*n^2 - 1600*n - 384)*x^2 - (512*n^2 + 1024*n)*x^3)*P[n - 1, x] + (25 + (100*n + 180)*x + (80*n^2 + 640*n + 368)*x^2 + (384*n^2 + 1088*n + 192)*x^3 + (256*n^2 + 512*n)*x^4)*D[P[n - 1, x], x])/(40*n - 40 + (80*n^2 + 48*n - 128)*x + (64*n^3 + 64*n^2 - 128*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(80*n - 200 + (240*n^2 - 464*n - 640)*x + (128*n^3 - 64*n^2 - 1600*n - 384)*x^2 - (512*n^2 + 1024*n)*x^3)*v[i - 1] + (25 + (100*n + 180)*x + (80*n^2 + 640*n + 368)*x^2 + (384*n^2 + 1088*n + 192)*x^3 + (256*n^2 + 512*n)*x^4)*deriv(v[i - 1], x))/(40*n - 40 + (80*n^2 + 48*n - 128)*x + (64*n^3 + 64*n^2 - 128*n)*x^2)); 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-392f49b496f79fbf","steps":["basis A013611","\u03b8+1"],"edges":["e-4d9d32144b57b963","e-33efebad88fe3ec5"],"text":"A026729 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A026729 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}