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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":"ac7098d0eb472975830e5dd569f5923baafcea59f0a554bb90024de5a2bfe5d3","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-9b5c7e97e5191da6","role":"sequence","name":"Triangular array formed by the Mersenne numbers.","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:    3,     4;\n  2:    7,    24,      9;\n  3:   15,    92,     81,     16;\n  4:   31,   288,    432,    192,     25;\n  5:   63,   804,   1782,   1312,    375,     36;\n  6:  127,  2088,   6291,   6720,   3125,    648,    49;\n  7:  255,  5164,  20007,  28720,  19125,   6372,  1029,    64;\n  8:  511, 12336,  59058, 108288,  96000,  45360, 11662,  1536,   81;\n  9: 1023, 28724, 164916, 372416, 418950, 261576, 94668, 19712, 2187, 100;\n","recurrence_sentence":"The row polynomials satisfy (8*n+6*n*x-(20*n^3-18*n)*x^2-30*n^3*x^3+2*n^5*x^4)*P(n,x) = (12*n+12+(7*n^2+25*n+18)*x-(30*n^3+33*n^2-39*n-42)*x^2-(2*n^4+62*n^3+77*n^2-13*n-30)*x^3+(3*n^5+3*n^4-27*n^3-54*n^2-33*n-6)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (1+9*x-(7*n^2+7*n-26)*x^2-(57*n^2+57*n-42)*x^3+(n^4+2*n^3-96*n^2-97*n-11)*x^4+(6*n^4+12*n^3-9*n^2-15*n-3)*x^5+(n^4+2*n^3+n^2)*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[((12*n + 12 + (7*n^2 + 25*n + 18)*x - (30*n^3 + 33*n^2 - 39*n - 42)*x^2 - (2*n^4 + 62*n^3 + 77*n^2 - 13*n - 30)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (1 + 9*x - (7*n^2 + 7*n - 26)*x^2 - (57*n^2 + 57*n - 42)*x^3 + (n^4 + 2*n^3 - 96*n^2 - 97*n - 11)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(8*n + 6*n*x - (20*n^3 - 18*n)*x^2 - 30*n^3*x^3 + 2*n^5*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] = ((12*n + 12 + (7*n^2 + 25*n + 18)*x - (30*n^3 + 33*n^2 - 39*n - 42)*x^2 - (2*n^4 + 62*n^3 + 77*n^2 - 13*n - 30)*x^3 + (3*n^5 + 3*n^4 - 27*n^3 - 54*n^2 - 33*n - 6)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (1 + 9*x - (7*n^2 + 7*n - 26)*x^2 - (57*n^2 + 57*n - 42)*x^3 + (n^4 + 2*n^3 - 96*n^2 - 97*n - 11)*x^4 + (6*n^4 + 12*n^3 - 9*n^2 - 15*n - 3)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(8*n + 6*n*x - (20*n^3 - 18*n)*x^2 - 30*n^3*x^3 + 2*n^5*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-9b5c7e97e5191da6","steps":["(\u03b8+1)\u00b2"],"edges":["e-f01a2d79bc5e0611"],"text":"A110441 \u2192 (\u03b8+1)\u00b2"},"headline":"A110441 \u2192 (\u03b8+1)\u00b2","chain":null}