{"id":"n-8e8b7bce030d60e7","revision":0,"first_index":0,"fingerprint":"f6-78732396ad9c62f4","name":"theta_plus_one of Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by 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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"c64ca521ca70ad621ad8b23a8b9ba059774d5c86bebd3e65d2901ba72b3a189e","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-14be0ce6edbde2a1","role":"sequence","name":"Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.","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: 1,    16,       3;\n  3: 1,    62,      93,         4;\n  4: 1,   204,     978,       408,         5;\n  5: 1,   634,    7218,      9624,      1585,         6;\n  6: 1,  1928,   45261,    137344,     75435,      5784,         7;\n  7: 1,  5814,  260019,   1523788,   1904735,    520038,     20349,        8;\n  8: 1, 17476, 1419252,  14600824,  34628590,  21901236,   3311588,    69904,      9;\n  9: 1, 52466, 7506912, 127499520, 516228110, 619473732, 223124160, 20018432, 236097, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n-4)*x^2*P(n-4,x) + ((8*n-4)*x-(16*n^2-32*n+4)*x^2)*P(n-3,x) - ((8*n-4)*x^2-(8*n-4)*x^3)*d/dx P(n-3,x) + (2*n-1+(8*n^2-8*n-6)*x-(8*n^3-20*n^2+6*n+9)*x^2)*P(n-2,x) + ((4*n-2)*x-(8*n^2-4*n-8)*x^2+(8*n^2-8*n-6)*x^3)*d/dx P(n-2,x) - (2-(8*n^2-16*n+6)*x)*P(n-1,x) + ((4*n-6)*x-(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 16*x + 3*x^2, P(3,x) = 1 + 62*x + 93*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 16*x + 3*x^2;\nP[3, x_] := 1 + 62*x + 93*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 4)*x^2*P[n - 4, x] + ((8*n - 4)*x - (16*n^2 - 32*n + 4)*x^2)*P[n - 3, x] - ((8*n - 4)*x^2 - (8*n - 4)*x^3)*D[P[n - 3, x], x] + (2*n - 1 + (8*n^2 - 8*n - 6)*x - (8*n^3 - 20*n^2 + 6*n + 9)*x^2)*P[n - 2, x] + ((4*n - 2)*x - (8*n^2 - 4*n - 8)*x^2 + (8*n^2 - 8*n - 6)*x^3)*D[P[n - 2, x], x] - (2 - (8*n^2 - 16*n + 6)*x)*P[n - 1, x] + ((4*n - 6)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 16*x + 3*x^2; v[4] = 1 + 62*x + 93*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(8*n - 4)*x^2*v[i - 4] + ((8*n - 4)*x - (16*n^2 - 32*n + 4)*x^2)*v[i - 3] - ((8*n - 4)*x^2 - (8*n - 4)*x^3)*deriv(v[i - 3], x) + (2*n - 1 + (8*n^2 - 8*n - 6)*x - (8*n^3 - 20*n^2 + 6*n + 9)*x^2)*v[i - 2] + ((4*n - 2)*x - (8*n^2 - 4*n - 8)*x^2 + (8*n^2 - 8*n - 6)*x^3)*deriv(v[i - 2], x) - (2 - (8*n^2 - 16*n + 6)*x)*v[i - 1] + ((4*n - 6)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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-14be0ce6edbde2a1","steps":["\u03b8+1"],"edges":["e-f96a1844c4c11a96"],"text":"A144439 \u2192 \u03b8+1"},"headline":"A144439 \u2192 \u03b8+1","chain":null}