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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":"d588ffb11af27c0949fad94f9842b3049b41965ed06ee2644d3ed52a79f9c9b8","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-cbf81d8f5ae9d341","role":"sequence","name":"veronese_section_2_0 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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: 4,   4;\n  1: 4,  20;\n  2: 4,  52,    9;\n  3: 4, 100,   63;\n  4: 4, 164,  234,    16;\n  5: 4, 244,  630,   144;\n  6: 4, 340, 1395,   688,   25;\n  7: 4, 452, 2709,  2352,  275;\n  8: 4, 580, 4788,  6496, 1600,  36;\n  9: 4, 724, 7884, 15456, 6600, 468;\n","recurrence_sentence":"The row polynomials satisfy (576*n-(352*n^3+672*n^2+80*n+1200)*x+(16*n^5+96*n^4+336*n^3+1000*n^2+1528*n+480)*x^2-(8*n^5+76*n^4+298*n^3+652*n^2+826*n+444)*x^3+(n^5+13*n^4+67*n^3+171*n^2+216*n+108)*x^4)*P(n,x) = (576*n-(352*n^3+672*n^2-2224*n+2352)*x+(16*n^5+96*n^4-272*n^3-968*n^2+1224*n-896)*x^2+(8*n^5+76*n^4+422*n^3+1280*n^2+1478*n+288)*x^3-(7*n^5+85*n^4+403*n^3+943*n^2+1090*n+448)*x^4+(n^5+15*n^4+87*n^3+241*n^2+312*n+144)*x^5)*P(n-1,x) + (1152*x-(704*n^2+2752*n+2656)*x^2+(32*n^4+320*n^3+2080*n^2+5728*n+4064)*x^3-(48*n^4+512*n^3+2244*n^2+4748*n+3968)*x^4+(18*n^4+220*n^3+1014*n^2+2108*n+1696)*x^5-(2*n^4+28*n^3+146*n^2+336*n+288)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 4*x.","mathematica":"P[0, x_] := 4 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((576*n - (352*n^3 + 672*n^2 - 2224*n + 2352)*x + (16*n^5 + 96*n^4 - 272*n^3 - 968*n^2 + 1224*n - 896)*x^2 + (8*n^5 + 76*n^4 + 422*n^3 + 1280*n^2 + 1478*n + 288)*x^3 - (7*n^5 + 85*n^4 + 403*n^3 + 943*n^2 + 1090*n + 448)*x^4 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^5)*P[n - 1, x] + (1152*x - (704*n^2 + 2752*n + 2656)*x^2 + (32*n^4 + 320*n^3 + 2080*n^2 + 5728*n + 4064)*x^3 - (48*n^4 + 512*n^3 + 2244*n^2 + 4748*n + 3968)*x^4 + (18*n^4 + 220*n^3 + 1014*n^2 + 2108*n + 1696)*x^5 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^6)*D[P[n - 1, x], x])/(576*n - (352*n^3 + 672*n^2 + 80*n + 1200)*x + (16*n^5 + 96*n^4 + 336*n^3 + 1000*n^2 + 1528*n + 480)*x^2 - (8*n^5 + 76*n^4 + 298*n^3 + 652*n^2 + 826*n + 444)*x^3 + (n^5 + 13*n^4 + 67*n^3 + 171*n^2 + 216*n + 108)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((576*n - (352*n^3 + 672*n^2 - 2224*n + 2352)*x + (16*n^5 + 96*n^4 - 272*n^3 - 968*n^2 + 1224*n - 896)*x^2 + (8*n^5 + 76*n^4 + 422*n^3 + 1280*n^2 + 1478*n + 288)*x^3 - (7*n^5 + 85*n^4 + 403*n^3 + 943*n^2 + 1090*n + 448)*x^4 + (n^5 + 15*n^4 + 87*n^3 + 241*n^2 + 312*n + 144)*x^5)*v[i - 1] + (1152*x - (704*n^2 + 2752*n + 2656)*x^2 + (32*n^4 + 320*n^3 + 2080*n^2 + 5728*n + 4064)*x^3 - (48*n^4 + 512*n^3 + 2244*n^2 + 4748*n + 3968)*x^4 + (18*n^4 + 220*n^3 + 1014*n^2 + 2108*n + 1696)*x^5 - (2*n^4 + 28*n^3 + 146*n^2 + 336*n + 288)*x^6)*deriv(v[i - 1], x))/(576*n - (352*n^3 + 672*n^2 + 80*n + 1200)*x + (16*n^5 + 96*n^4 + 336*n^3 + 1000*n^2 + 1528*n + 480)*x^2 - (8*n^5 + 76*n^4 + 298*n^3 + 652*n^2 + 826*n + 444)*x^3 + (n^5 + 13*n^4 + 67*n^3 + 171*n^2 + 216*n + 108)*x^4)); 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-f48bed27dbc29c5f","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-9f9d452726caee29","e-0f9d56035b7ea712"],"text":"A391533 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A391533 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}