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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"194a4a8fd8954c7b548db292b5c27fc067ea6649190dbd6659eaaf5326a90c77","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-bbfc5a54ff13a785","role":"sequence","name":"Triangle of coefficients in expansion of (1+5x)^n.","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,  40,   225;\n  1: 1,  60,   675,   2000;\n  2: 1,  80,  1350,   8000,   15625;\n  3: 1, 100,  2250,  20000,   78125,   112500;\n  4: 1, 120,  3375,  40000,  234375,   675000,    765625;\n  5: 1, 140,  4725,  70000,  546875,  2362500,   5359375,    5000000;\n  6: 1, 160,  6300, 112000, 1093750,  6300000,  21437500,   40000000,   31640625;\n  7: 1, 180,  8100, 168000, 1968750, 14175000,  64312500,  180000000,  284765625,   195312500;\n  8: 1, 200, 10125, 240000, 3281250, 28350000, 160781250,  600000000, 1423828125,  1953125000,  1181640625;\n  9: 1, 220, 12375, 330000, 5156250, 51975000, 353718750, 1650000000, 5220703125, 10742187500, 12998046875, 7031250000;\n","recurrence_sentence":"The row polynomials satisfy (1+(15*n+25)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (1+(15*n+45)*x+(25*n^2+225*n+425)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 40*x + 225*x^2.","mathematica":"P[0, x_] := 1 + 40*x + 225*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (15*n + 45)*x + (25*n^2 + 225*n + 425)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(1 + (15*n + 25)*x + (25*n^2 + 100*n + 100)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 40*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (15*n + 45)*x + (25*n^2 + 225*n + 425)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(1 + (15*n + 25)*x + (25*n^2 + 100*n + 100)*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-bbfc5a54ff13a785","steps":["(\u03b8+1)\u00b2"],"edges":["e-0fd3160d73fec8b0"],"text":"A013612 \u2192 (\u03b8+1)\u00b2"},"headline":"A013612 \u2192 (\u03b8+1)\u00b2","chain":null}