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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":"ded5d923da65ff6e916b0e055d4d4faea91904be175dba2d0cb68889ed3dc882","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-8588fa6ee4014167","role":"sequence","name":"basis_map_A013611 of Riordan array (1, 2+2x).","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,     16;\n  2:    6,     80,     192;\n  3:   16,    320,    1536,     2048;\n  4:   44,   1152,    8256,    22528,     20480;\n  5:  120,   3904,   37248,   153600,    286720,     196608;\n  6:  328,  12736,  151680,   837632,   2375680,    3342336,    1835008;\n  7:  896,  40448,  577536,  3997696,  15237120,   32636928,   36700160,   16777216;\n  8: 2448, 125952, 2096640, 17448960,  83578880,  241041408,  412876800,  385875968,  150994944;\n  9: 6688, 386304, 7345152, 71417856, 412098560, 1495990272, 3453485056, 4915724288, 3925868544, 1342177280;\n","recurrence_sentence":"The row polynomials satisfy (12*n+(24*n^2+64*n)*x+(16*n^3+80*n^2+64*n)*x^2)*P(n,x) = -(24*n-48+(72*n^2-40*n-400)*x+(32*n^3+96*n^2-768*n-1088)*x^2-(256*n^2+1280*n+1024)*x^3)*P(n-1,x) + (9+(36*n+108)*x+(24*n^2+312*n+464)*x^2+(128*n^2+832*n+832)*x^3+(128*n^2+640*n+512)*x^4)*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[(-(24*n - 48 + (72*n^2 - 40*n - 400)*x + (32*n^3 + 96*n^2 - 768*n - 1088)*x^2 - (256*n^2 + 1280*n + 1024)*x^3)*P[n - 1, x] + (9 + (36*n + 108)*x + (24*n^2 + 312*n + 464)*x^2 + (128*n^2 + 832*n + 832)*x^3 + (128*n^2 + 640*n + 512)*x^4)*D[P[n - 1, x], x])/(12*n + (24*n^2 + 64*n)*x + (16*n^3 + 80*n^2 + 64*n)*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] = (-(24*n - 48 + (72*n^2 - 40*n - 400)*x + (32*n^3 + 96*n^2 - 768*n - 1088)*x^2 - (256*n^2 + 1280*n + 1024)*x^3)*v[i - 1] + (9 + (36*n + 108)*x + (24*n^2 + 312*n + 464)*x^2 + (128*n^2 + 832*n + 832)*x^3 + (128*n^2 + 640*n + 512)*x^4)*deriv(v[i - 1], x))/(12*n + (24*n^2 + 64*n)*x + (16*n^3 + 80*n^2 + 64*n)*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-f5a001953ee28e8f","steps":["basis A013611","\u03b8+1"],"edges":["e-5d40644b540aa393","e-0f3e1ccf3d62ec50"],"text":"A099040 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A099040 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}