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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":"17e45f93c76ce84aead0857dbe84bfc8ef7644185226b3a55dc87791d9b6aa25","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-b4a66ef190cb2745","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: triangle of repeated stepped binomial coefficients.","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,   2;\n  2:  2,   6,    3;\n  3:  2,  10,   12,    4;\n  4:  4,  24,   39,   24,    5;\n  5:  4,  32,   75,   76,   35,    6;\n  6:  8,  72,  198,  252,  165,   54,    7;\n  7:  8,  88,  306,  516,  480,  252,   70,   8;\n  8: 16, 192,  744, 1440, 1605, 1080,  434,  96,   9;\n  9: 16, 224, 1032, 2432, 3405, 3006, 1694, 592, 117, 10;\n","recurrence_sentence":"The row polynomials satisfy (32+(24*n+52)*x+(9*n^2+29*n+20)*x^2+(12*n^2+2*n)*x^3+(4*n^2-4*n)*x^4)*P(n,x) = -(96*n-128+(108*n^2+96*n-344)*x+(27*n^3+195*n^2-128*n-304)*x^2+(54*n^3+91*n^2-201*n-74)*x^3+(36*n^3-12*n^2-74*n+22)*x^4+(8*n^3-12*n^2-4*n+8)*x^5)*P(n-1,x) + (64+(96*n+208)*x+(36*n^2+292*n+232)*x^2+(102*n^2+326*n+80)*x^3+(106*n^2+158*n-28)*x^4+(48*n^2+28*n-24)*x^5+(8*n^2-4)*x^6)*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[(-(96*n - 128 + (108*n^2 + 96*n - 344)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (54*n^3 + 91*n^2 - 201*n - 74)*x^3 + (36*n^3 - 12*n^2 - 74*n + 22)*x^4 + (8*n^3 - 12*n^2 - 4*n + 8)*x^5)*P[n - 1, x] + (64 + (96*n + 208)*x + (36*n^2 + 292*n + 232)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (106*n^2 + 158*n - 28)*x^4 + (48*n^2 + 28*n - 24)*x^5 + (8*n^2 - 4)*x^6)*D[P[n - 1, x], x])/(32 + (24*n + 52)*x + (9*n^2 + 29*n + 20)*x^2 + (12*n^2 + 2*n)*x^3 + (4*n^2 - 4*n)*x^4)]];\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] = (-(96*n - 128 + (108*n^2 + 96*n - 344)*x + (27*n^3 + 195*n^2 - 128*n - 304)*x^2 + (54*n^3 + 91*n^2 - 201*n - 74)*x^3 + (36*n^3 - 12*n^2 - 74*n + 22)*x^4 + (8*n^3 - 12*n^2 - 4*n + 8)*x^5)*v[i - 1] + (64 + (96*n + 208)*x + (36*n^2 + 292*n + 232)*x^2 + (102*n^2 + 326*n + 80)*x^3 + (106*n^2 + 158*n - 28)*x^4 + (48*n^2 + 28*n - 24)*x^5 + (8*n^2 - 4)*x^6)*deriv(v[i - 1], x))/(32 + (24*n + 52)*x + (9*n^2 + 29*n + 20)*x^2 + (12*n^2 + 2*n)*x^3 + (4*n^2 - 4*n)*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-35bd700768cac55f","steps":["basis A007318","\u03b8+1"],"edges":["e-01d2a4c51ed4f80c","e-af01f83cf1cbc74b"],"text":"A103633 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A103633 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}