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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":"fcf59ad4711b8309428eaa3a512a9273df1d39a291400e27b3d56f90c58bb6f1","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-8fd7b2e367875124","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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;\n   2:    3,      8;\n   3:    8,     48,      36;\n   4:   21,    200,     324,     128;\n   5:   55,    720,    1836,    1536,     400;\n   6:  144,   2400,    8424,   11008,    6000,     1152;\n   7:  377,   7632,   34236,   61440,   52000,    20736,    3136;\n   8:  987,  23512,  128628,  295040,  342000,   210816,   65856,    8192;\n   9: 2584,  70800,  457452, 1281024, 1896000,  1596672,  768320,  196608,  20736;\n  10: 6765, 209560, 1562328, 5174528, 9349200, 10039680, 6585600, 2588672, 559872, 51200;\n","recurrence_sentence":"The row polynomials satisfy (400*n-400+(120*n-120)*x-(800*n^3-2400*n^2+2256*n-656)*x^2-(480*n^3-1440*n^2+1440*n-480)*x^3+(64*n^5-320*n^4+640*n^3-640*n^2+320*n-64)*x^4)*P(n,x) = (600*n+(700*n^2+380*n)*x-(1200*n^3-2280*n^2+264*n)*x^2-(160*n^4+1440*n^3-2000*n^2+304*n)*x^3+(96*n^5-384*n^4-288*n^3+480*n^2-96*n)*x^4+(64*n^5-128*n^4+64*n^3)*x^5)*P(n-1,x) + (125+450*x-(700*n^2-700*n-440)*x^2-(2280*n^2-2280*n+48)*x^3+(80*n^4-160*n^3-1920*n^2+2000*n-304)*x^4+(192*n^4-384*n^3-288*n^2+480*n-96)*x^5+(64*n^4-128*n^3+64*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((600*n + (700*n^2 + 380*n)*x - (1200*n^3 - 2280*n^2 + 264*n)*x^2 - (160*n^4 + 1440*n^3 - 2000*n^2 + 304*n)*x^3 + (96*n^5 - 384*n^4 - 288*n^3 + 480*n^2 - 96*n)*x^4 + (64*n^5 - 128*n^4 + 64*n^3)*x^5)*P[n - 1, x] + (125 + 450*x - (700*n^2 - 700*n - 440)*x^2 - (2280*n^2 - 2280*n + 48)*x^3 + (80*n^4 - 160*n^3 - 1920*n^2 + 2000*n - 304)*x^4 + (192*n^4 - 384*n^3 - 288*n^2 + 480*n - 96)*x^5 + (64*n^4 - 128*n^3 + 64*n^2)*x^6)*D[P[n - 1, x], x])/(400*n - 400 + (120*n - 120)*x - (800*n^3 - 2400*n^2 + 2256*n - 656)*x^2 - (480*n^3 - 1440*n^2 + 1440*n - 480)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((600*n + (700*n^2 + 380*n)*x - (1200*n^3 - 2280*n^2 + 264*n)*x^2 - (160*n^4 + 1440*n^3 - 2000*n^2 + 304*n)*x^3 + (96*n^5 - 384*n^4 - 288*n^3 + 480*n^2 - 96*n)*x^4 + (64*n^5 - 128*n^4 + 64*n^3)*x^5)*v[i - 1] + (125 + 450*x - (700*n^2 - 700*n - 440)*x^2 - (2280*n^2 - 2280*n + 48)*x^3 + (80*n^4 - 160*n^3 - 1920*n^2 + 2000*n - 304)*x^4 + (192*n^4 - 384*n^3 - 288*n^2 + 480*n - 96)*x^5 + (64*n^4 - 128*n^3 + 64*n^2)*x^6)*deriv(v[i - 1], x))/(400*n - 400 + (120*n - 120)*x - (800*n^3 - 2400*n^2 + 2256*n - 656)*x^2 - (480*n^3 - 1440*n^2 + 1440*n - 480)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*x^4)); 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-976cdb6e4a693ec8","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-567d6f79aa6a87e5","e-c87207b6cb14ea0b"],"text":"A078812 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A078812 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}