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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":"887ff0365a3cc3abc618480d51381582c0b38a7195c7b389efe148c62a16f7ab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c1716beb5dbd6d37","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,       4;\n   3:      8,      32,       9;\n   4:     32,     192,     162,       16;\n   5:    128,    1024,    1440,      512,       25;\n   6:    512,    5120,   10080,     6400,     1250,      36;\n   7:   2048,   24576,   62208,    57344,    21000,    2592,      49;\n   8:   8192,  114688,  354816,   430080,   235200,   56448,    4802,     64;\n   9:  32768,  524288, 1916928,  2883584,  2112000,  774144,  131712,   8192,    81;\n  10: 131072, 2359296, 9953280, 17891328, 16473600, 8211456, 2173248, 276480, 13122, 100;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-80*n^2+132*n-72+(24*n^4-156*n^3+376*n^2-398*n+156)*x+(4*n^5-28*n^4+77*n^3-104*n^2+69*n-18)*x^2)*P(n,x) = (64*n^3-320*n^2+528*n-288+(96*n^4-432*n^3+544*n^2-8*n-240)*x+(16*n^5+8*n^4-324*n^3+726*n^2-496*n+40)*x^2+(12*n^5-52*n^4+63*n^3-30*n+4)*x^3)*P(n-1,x) - ((32*n^2-96*n+72)*x+(48*n^3-112*n^2+16*n+68)*x^2+(8*n^4+32*n^3-146*n^2+126*n-6)*x^3+(8*n^4-16*n^3-2*n^2+14*n-2)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((64*n^3 - 320*n^2 + 528*n - 288 + (96*n^4 - 432*n^3 + 544*n^2 - 8*n - 240)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*P[n - 1, x] - ((32*n^2 - 96*n + 72)*x + (48*n^3 - 112*n^2 + 16*n + 68)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*D[P[n - 1, x], x])/(16*n^3 - 80*n^2 + 132*n - 72 + (24*n^4 - 156*n^3 + 376*n^2 - 398*n + 156)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((64*n^3 - 320*n^2 + 528*n - 288 + (96*n^4 - 432*n^3 + 544*n^2 - 8*n - 240)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (12*n^5 - 52*n^4 + 63*n^3 - 30*n + 4)*x^3)*v[i - 1] - ((32*n^2 - 96*n + 72)*x + (48*n^3 - 112*n^2 + 16*n + 68)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (8*n^4 - 16*n^3 - 2*n^2 + 14*n - 2)*x^4)*deriv(v[i - 1], x))/(16*n^3 - 80*n^2 + 132*n - 72 + (24*n^4 - 156*n^3 + 376*n^2 - 398*n + 156)*x + (4*n^5 - 28*n^4 + 77*n^3 - 104*n^2 + 69*n - 18)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-219ad3693217b36c","e-b4aa1f3c9560d590"],"text":"A086645 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A086645 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}