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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":"525c4faa8673766f3e0505d553ed668faa63f04783578eb58c22a5498e207e2c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6a8ffbe72ba42678","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.","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,    40,       9;\n   2: 1,    92,     207,        16;\n   3: 1,   200,    1242,       800,         25;\n   4: 1,   420,    5526,      9824,       2625,          36;\n   5: 1,   864,   21303,     78592,      59175,        7776,          49;\n   6: 1,  1756,   75573,    503728,     787075,      302292,       21511,         64;\n   7: 1,  3544,  254376,   2821024,    7870750,     6347304,     1384936,      56704,         81;\n   8: 1,  7124,  826920,  14449664,   65800850,    94753224,    44252096,    5880320,     144261,      100;\n   9: 1, 14288, 2625021,  69559296,  487047450,  1137038688,   954613002,  278237184,   23625189,   357200,    121;\n  10: 1, 28620, 8196543, 320237520, 3304728450, 11731670424, 15968106966, 8460104832, 1621202445, 91072700, 865755, 144;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -((4*n^2+8*n)*x-(9*n^3+25*n^2+16*n)*x^2)*P(n-2,x) - ((2*n+4)*x-(4*n^2+16*n+17)*x^2-(4*n^3-2*n^2-29*n-23)*x^3)*d/dx P(n-2,x) - ((n+2)*x^2-(4*n^2+12*n+9)*x^3+(4*n^2+11*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (n+1+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+3)*x-(4*n^2+12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 40*x + 9*x^2, P(2,x) = 1 + 92*x + 207*x^2 + 16*x^3.","mathematica":"P[1, x_] := 1 + 40*x + 9*x^2;\nP[2, x_] := 1 + 92*x + 207*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*P[n - 2, x] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 1 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 40*x + 9*x^2; v[2] = 1 + 92*x + 207*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*v[i - 2] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 1 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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-6a8ffbe72ba42678","steps":["(\u03b8+1)\u00b2"],"edges":["e-22dd3938cdb718b1"],"text":"A166341 \u2192 (\u03b8+1)\u00b2"},"headline":"A166341 \u2192 (\u03b8+1)\u00b2","chain":null}