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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":"b31d69715a4ade4271c5ebed781eee60a64bc941e7e98a65055d5201e8946f52","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2b477f2aa77e3efc","role":"sequence","name":"Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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: 1,   4;\n  2: 1,  16,    27;\n  3: 1,  36,   207,    240;\n  4: 1,  64,   774,   2816,     2625;\n  5: 1, 100,  2070,  15200,    42225,     34020;\n  6: 1, 144,  4545,  55680,   303475,    702864,     509355;\n  7: 1, 196,  8757, 160720,  1434475,   6383916,   12946143,    8648640;\n  8: 1, 256, 15372, 394240,  5201350,  37368576,  143284428,  262287360,  164189025;\n  9: 1, 324, 25164, 858816, 15673350, 164697624, 1007951756, 3443786496, 5807465505, 3445942500;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(2*n-1+(16*n^2-32*n+12)*x-(72*n^3-252*n^2+270*n-81)*x^2)*P(n-2,x) + ((16*n^2-32*n+12)*x^2+(32*n^3-112*n^2+120*n-36)*x^3)*d/dx P(n-2,x) + (4*n-4+(16*n^2-32*n+12)*x)*P(n-1,x) - (16*n^2-32*n+12)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (16*n^2 - 32*n + 12)*x - (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*P[n - 2, x] + ((16*n^2 - 32*n + 12)*x^2 + (32*n^3 - 112*n^2 + 120*n - 36)*x^3)*D[P[n - 2, x], x] + (4*n - 4 + (16*n^2 - 32*n + 12)*x)*P[n - 1, x] - (16*n^2 - 32*n + 12)*x^2*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 1 + (16*n^2 - 32*n + 12)*x - (72*n^3 - 252*n^2 + 270*n - 81)*x^2)*v[i - 2] + ((16*n^2 - 32*n + 12)*x^2 + (32*n^3 - 112*n^2 + 120*n - 36)*x^3)*deriv(v[i - 2], x) + (4*n - 4 + (16*n^2 - 32*n + 12)*x)*v[i - 1] - (16*n^2 - 32*n + 12)*x^2*deriv(v[i - 1], x))/(2*n - 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-2b477f2aa77e3efc","steps":["(\u03b8+1)\u00b2"],"edges":["e-3a71cfd7d0b9ec97"],"text":"A109692 \u2192 (\u03b8+1)\u00b2"},"headline":"A109692 \u2192 (\u03b8+1)\u00b2","chain":null}