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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":"bb6b659df9a425fadf740ec27824f3feddf989643f18010ddc85c35d95da7632","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-0836bf76d9ba1ba5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(k,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:  2,    4;\n  2:  3,   16,    12;\n  3:  5,   40,    72,    32;\n  4:  8,   88,   252,   256,     80;\n  5: 13,  180,   708,  1152,    800,    192;\n  6: 21,  352,  1764,  3968,   4400,   2304,    448;\n  7: 34,  668,  4068, 11712,  18000,  14976,   6272,   1024;\n  8: 55, 1240,  8892, 31232,  61600,  71040,  47040,  16384,  2304;\n  9: 89, 2264, 18684, 77504, 186960, 277056, 254016, 139264, 41472, 5120;\n","recurrence_sentence":"The row polynomials satisfy (20*n+20+(20*n^2+52*n+32)*x+(8*n^3+16*n^2+8*n)*x^2-(16*n^2+16*n)*x^3)*P(n,x) = -(40*n-120+(60*n^2-156*n-272)*x+(16*n^3-32*n^2-288*n-176)*x^2-(80*n^2+48*n-64)*x^3+(64*n+64)*x^4)*P(n-1,x) + (25+(50*n+90)*x+(20*n^2+140*n+72)*x^2+(48*n^2+48*n-64)*x^3+(16*n^2-80*n-112)*x^4-(32*n+32)*x^5)*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[(-(40*n - 120 + (60*n^2 - 156*n - 272)*x + (16*n^3 - 32*n^2 - 288*n - 176)*x^2 - (80*n^2 + 48*n - 64)*x^3 + (64*n + 64)*x^4)*P[n - 1, x] + (25 + (50*n + 90)*x + (20*n^2 + 140*n + 72)*x^2 + (48*n^2 + 48*n - 64)*x^3 + (16*n^2 - 80*n - 112)*x^4 - (32*n + 32)*x^5)*D[P[n - 1, x], x])/(20*n + 20 + (20*n^2 + 52*n + 32)*x + (8*n^3 + 16*n^2 + 8*n)*x^2 - (16*n^2 + 16*n)*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] = (-(40*n - 120 + (60*n^2 - 156*n - 272)*x + (16*n^3 - 32*n^2 - 288*n - 176)*x^2 - (80*n^2 + 48*n - 64)*x^3 + (64*n + 64)*x^4)*v[i - 1] + (25 + (50*n + 90)*x + (20*n^2 + 140*n + 72)*x^2 + (48*n^2 + 48*n - 64)*x^3 + (16*n^2 - 80*n - 112)*x^4 - (32*n + 32)*x^5)*deriv(v[i - 1], x))/(20*n + 20 + (20*n^2 + 52*n + 32)*x + (8*n^3 + 16*n^2 + 8*n)*x^2 - (16*n^2 + 16*n)*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-21854f4257effb01","steps":["basis A013609","\u03b8+1"],"edges":["e-38aa524f6e49a0d8","e-07eeeb9089ebc8b6"],"text":"A030528 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A030528 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}