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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":"5ca3b6fdf769e3928b10797f9f5d34cc46591e7200a08cf5f68daeaad328cf68","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-7c7a913d07444057","role":"sequence","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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,      8;\n  2:    8,     40,     36;\n  3:   21,    152,    264,     128;\n  4:   55,    520,   1332,    1280,     400;\n  5:  144,   1680,   5688,    8192,    5200,    1152;\n  6:  377,   5232,  22032,   42368,   40880,   18816,    3136;\n  7:  987,  15880,  79992,  192640,  250400,  177408,   62720,    8192;\n  8: 2584,  47288, 277308,  802816, 1314800, 1264896,  696192,  196608,  20736;\n  9: 6765, 138760, 928536, 3141120, 6211600, 7570560, 5698560, 2531328, 587520, 51200;\n","recurrence_sentence":"The row polynomials satisfy (40*n+(80*n^2+56*n)*x+(64*n^3+32*n^2)*x^2)*P(n,x) = (160*n-40+(400*n^2+504*n-176)*x+(256*n^3+1024*n^2+320*n-224)*x^2+(384*n^3+576*n^2-128*n-64)*x^3)*P(n-1,x) - (25+(100*n+160)*x+(80*n^2+520*n+372)*x^2+(288*n^2+896*n+368)*x^3+(256*n^2+512*n+128)*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[((160*n - 40 + (400*n^2 + 504*n - 176)*x + (256*n^3 + 1024*n^2 + 320*n - 224)*x^2 + (384*n^3 + 576*n^2 - 128*n - 64)*x^3)*P[n - 1, x] - (25 + (100*n + 160)*x + (80*n^2 + 520*n + 372)*x^2 + (288*n^2 + 896*n + 368)*x^3 + (256*n^2 + 512*n + 128)*x^4)*D[P[n - 1, x], x])/(40*n + (80*n^2 + 56*n)*x + (64*n^3 + 32*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] = ((160*n - 40 + (400*n^2 + 504*n - 176)*x + (256*n^3 + 1024*n^2 + 320*n - 224)*x^2 + (384*n^3 + 576*n^2 - 128*n - 64)*x^3)*v[i - 1] - (25 + (100*n + 160)*x + (80*n^2 + 520*n + 372)*x^2 + (288*n^2 + 896*n + 368)*x^3 + (256*n^2 + 512*n + 128)*x^4)*deriv(v[i - 1], x))/(40*n + (80*n^2 + 56*n)*x + (64*n^3 + 32*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-79ebafb266da3b41","steps":["basis A013609","\u03b8+1"],"edges":["e-1a56a7670c2d3c41","e-ef82731e881bc045"],"text":"A172431 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A172431 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}