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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":"3b662c84769ae9ffb000964c81acf44697cd115c1244cd9113a3213e38baac14","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-6758fe7729845be8","role":"sequence","name":"Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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, 12,    9;\n  3: 1, 20,   54,   16;\n  4: 1, 28,  135,  160,    25;\n  5: 1, 36,  252,  560,   375,    36;\n  6: 1, 44,  405, 1344,  1750,   756,    49;\n  7: 1, 52,  594, 2640,  5250,  4536,  1372,    64;\n  8: 1, 60,  819, 4576, 12375, 16632, 10290,  2304,   81;\n  9: 1, 68, 1080, 7280, 25025, 46332, 45276, 21120, 3645, 100;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-4*n^2+(48*n^4-72*n^3+28*n^2-2*n)*x+(32*n^5-48*n^4+24*n^3-4*n^2)*x^2)*P(n,x) = (8*n^3-4*n^2+(48*n^4+24*n^3-52*n^2-2*n)*x+(32*n^5+192*n^4-120*n^3-152*n^2+48*n+12)*x^2+(96*n^5+80*n^4-152*n^3-52*n^2+56*n-12)*x^3)*P(n-1,x) - (4*n^2*x+(24*n^3+28*n^2-6*n-1)*x^2+(16*n^4+128*n^3+52*n^2-32*n-2)*x^3+(64*n^4+128*n^3+16*n^2-32*n+8)*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[((8*n^3 - 4*n^2 + (48*n^4 + 24*n^3 - 52*n^2 - 2*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^3)*P[n - 1, x] - (4*n^2*x + (24*n^3 + 28*n^2 - 6*n - 1)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^4)*D[P[n - 1, x], x])/(8*n^3 - 4*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (32*n^5 - 48*n^4 + 24*n^3 - 4*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] = ((8*n^3 - 4*n^2 + (48*n^4 + 24*n^3 - 52*n^2 - 2*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^3)*v[i - 1] - (4*n^2*x + (24*n^3 + 28*n^2 - 6*n - 1)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^4)*deriv(v[i - 1], x))/(8*n^3 - 4*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (32*n^5 - 48*n^4 + 24*n^3 - 4*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-6758fe7729845be8","steps":["(\u03b8+1)\u00b2"],"edges":["e-34cb5f7197630da3"],"text":"A054142 \u2192 (\u03b8+1)\u00b2"},"headline":"A054142 \u2192 (\u03b8+1)\u00b2","chain":null}