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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":"dc23553907a8c95491293256ee89bdb88ad91ed2fde091a1adacb76a9ba3e343","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-b3bdd177c3ec2fb9","role":"sequence","name":"Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.","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,    2;\n  1: 1,    6,      9,       4;\n  2: 1,   14,     48,      64,       35,         6;\n  3: 1,   30,    183,     460,      575,       366,       105,          8;\n  4: 1,   62,    618,    2504,     5280,      6336,      4382,       1648,        279,         10;\n  5: 1,  126,   1977,   11956,    37770,     70644,     82418,      60432,      26901,       6590,        693,         12;\n  6: 1,  254,   6156,   53232,   236745,    630594,   1082368,    1236992,     945891,     473490,     146388,      24624,       1651,         14;\n  7: 1,  510,  18891,  227740,  1373385,   4945986,  11626699,   18700056,   21037563,   16609570,    9067641,    3296124,     740155,      88158,      3825,        16;\n  8: 1, 1022,  57486,  951448,  7592390,  35764908, 110230078,  235429232,  359861760,  399846400,  323715194,  188965848,   77490634,   21258692,   3567930,    306592,     8687,      18;\n  9: 1, 2046, 174045, 3917188, 40684220, 245107896, 962147116, 2629336160, 5204306358, 7635462340, 8399008574, 6939075144, 4272671260, 1924294232, 612769740, 130189504, 16648049, 1044270, 19437, 20;\n","recurrence_sentence":"The row polynomials satisfy (1+(4*n-2)*x)*P(n,x) = ((2*n-2)*x^2+(8*n^2-8*n)*x^3-(20*n^2-14*n-6)*x^4-(16*n^3-16*n^2-4*n+4)*x^5)*P(n-2,x) - (x^2+(6*n+1)*x^3+(8*n^2-3)*x^4-(6*n+1)*x^5-(8*n^2-2)*x^6)*d/dx P(n-2,x) + (1+4*n*x+(12*n-3)*x^2+(16*n^2-12*n+2)*x^3)*P(n-1,x) + (x+(4*n-2)*x^2-x^3-(4*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 2*x, P(1,x) = 1 + 6*x + 9*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1 + 2*x;\nP[1, x_] := 1 + 6*x + 9*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n - 2)*x^2 + (8*n^2 - 8*n)*x^3 - (20*n^2 - 14*n - 6)*x^4 - (16*n^3 - 16*n^2 - 4*n + 4)*x^5)*P[n - 2, x] - (x^2 + (6*n + 1)*x^3 + (8*n^2 - 3)*x^4 - (6*n + 1)*x^5 - (8*n^2 - 2)*x^6)*D[P[n - 2, x], x] + (1 + 4*n*x + (12*n - 3)*x^2 + (16*n^2 - 12*n + 2)*x^3)*P[n - 1, x] + (x + (4*n - 2)*x^2 - x^3 - (4*n - 2)*x^4)*D[P[n - 1, x], x])/(1 + (4*n - 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; v[2] = 1 + 6*x + 9*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 2)*x^2 + (8*n^2 - 8*n)*x^3 - (20*n^2 - 14*n - 6)*x^4 - (16*n^3 - 16*n^2 - 4*n + 4)*x^5)*v[i - 2] - (x^2 + (6*n + 1)*x^3 + (8*n^2 - 3)*x^4 - (6*n + 1)*x^5 - (8*n^2 - 2)*x^6)*deriv(v[i - 2], x) + (1 + 4*n*x + (12*n - 3)*x^2 + (16*n^2 - 12*n + 2)*x^3)*v[i - 1] + (x + (4*n - 2)*x^2 - x^3 - (4*n - 2)*x^4)*deriv(v[i - 1], x))/(1 + (4*n - 2)*x)); 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":null,"headline":"theta_plus_one of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.","chain":null}