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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":"e5c6d654f76a2905a15896854f8cb542f49b0d129fbf588b6a364a342ab21ce5","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-cc665cf2fd1f1fd7","role":"sequence","name":"gamma_polynomial of Pascal-(1,4,1) array.","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;\n  2: 1,  16;\n  3: 1,  32;\n  4: 1,  48,  144;\n  5: 1,  64,  432;\n  6: 1,  80,  864,  1024;\n  7: 1,  96, 1440,  4096;\n  8: 1, 112, 2160, 10240,  6400;\n  9: 1, 128, 3024, 20480, 32000;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n+(12*n^3-8*n)*x+(16*n^4+16*n^3)*x^2)*P(n,x) = (n^2+n+(12*n^3+32*n^2-8*n-32)*x+(16*n^4+176*n^3+128*n^2-224*n-64)*x^2+(128*n^4+512*n^3+384*n^2-512*n-512)*x^3)*P(n-1,x) - ((n+1)*x+(12*n^2+40*n+24)*x^2+(16*n^3+272*n^2+512*n+192)*x^3+(256*n^3+1280*n^2+2048*n+1024)*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[((n^2 + n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (16*n^4 + 176*n^3 + 128*n^2 - 224*n - 64)*x^2 + (128*n^4 + 512*n^3 + 384*n^2 - 512*n - 512)*x^3)*P[n - 1, x] - ((n + 1)*x + (12*n^2 + 40*n + 24)*x^2 + (16*n^3 + 272*n^2 + 512*n + 192)*x^3 + (256*n^3 + 1280*n^2 + 2048*n + 1024)*x^4)*D[P[n - 1, x], x])/(n^2 + n + (12*n^3 - 8*n)*x + (16*n^4 + 16*n^3)*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] = ((n^2 + n + (12*n^3 + 32*n^2 - 8*n - 32)*x + (16*n^4 + 176*n^3 + 128*n^2 - 224*n - 64)*x^2 + (128*n^4 + 512*n^3 + 384*n^2 - 512*n - 512)*x^3)*v[i - 1] - ((n + 1)*x + (12*n^2 + 40*n + 24)*x^2 + (16*n^3 + 272*n^2 + 512*n + 192)*x^3 + (256*n^3 + 1280*n^2 + 2048*n + 1024)*x^4)*deriv(v[i - 1], x))/(n^2 + n + (12*n^3 - 8*n)*x + (16*n^4 + 16*n^3)*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-ed9434bfd889c409","steps":["\u03b3","(\u03b8+1)\u00b2"],"edges":["e-5cd8190965f3efc9","e-f5acb067356a2d03"],"text":"A081579 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2"},"headline":"A081579 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2","chain":null}