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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":"e3f7a731a49300f1c02da54f9f03cadcd451b2d67d7773dde6a9e823cb47e32d","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-91f141c4f7274be3","role":"sequence","name":"basis_map_A013609 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,       8;\n   3:      8,      64,       36;\n   4:     32,     384,      648,       128;\n   5:    128,    2048,     5760,      4096,       400;\n   6:    512,   10240,    40320,     51200,     20000,      1152;\n   7:   2048,   49152,   248832,    458752,    336000,     82944,      3136;\n   8:   8192,  229376,  1419264,   3440640,   3763200,   1806336,    307328,     8192;\n   9:  32768, 1048576,  7667712,  23068672,  33792000,  24772608,   8429568,  1048576,   20736;\n  10: 131072, 4718592, 39813120, 143130624, 263577600, 262766592, 139087872, 35389440, 3359232, 51200;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-20*n^2+33*n-18+(12*n^4-78*n^3+188*n^2-199*n+78)*x+(4*n^5-28*n^4+77*n^3-104*n^2+69*n-18)*x^2)*P(n,x) = (16*n^3-80*n^2+132*n-72+(48*n^4-216*n^3+272*n^2-4*n-120)*x+(16*n^5+8*n^4-324*n^3+726*n^2-496*n+40)*x^2+(24*n^5-104*n^4+126*n^3-60*n+8)*x^3)*P(n-1,x) - ((8*n^2-24*n+18)*x+(24*n^3-56*n^2+8*n+34)*x^2+(8*n^4+32*n^3-146*n^2+126*n-6)*x^3+(16*n^4-32*n^3-4*n^2+28*n-4)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 80*n^2 + 132*n - 72 + (48*n^4 - 216*n^3 + 272*n^2 - 4*n - 120)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (24*n^5 - 104*n^4 + 126*n^3 - 60*n + 8)*x^3)*P[n - 1, x] - ((8*n^2 - 24*n + 18)*x + (24*n^3 - 56*n^2 + 8*n + 34)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^3 - 20*n^2 + 33*n - 18 + (12*n^4 - 78*n^3 + 188*n^2 - 199*n + 78)*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 + 8*x; for(n = 3, N, my(i = n); v[i] = ((16*n^3 - 80*n^2 + 132*n - 72 + (48*n^4 - 216*n^3 + 272*n^2 - 4*n - 120)*x + (16*n^5 + 8*n^4 - 324*n^3 + 726*n^2 - 496*n + 40)*x^2 + (24*n^5 - 104*n^4 + 126*n^3 - 60*n + 8)*x^3)*v[i - 1] - ((8*n^2 - 24*n + 18)*x + (24*n^3 - 56*n^2 + 8*n + 34)*x^2 + (8*n^4 + 32*n^3 - 146*n^2 + 126*n - 6)*x^3 + (16*n^4 - 32*n^3 - 4*n^2 + 28*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^3 - 20*n^2 + 33*n - 18 + (12*n^4 - 78*n^3 + 188*n^2 - 199*n + 78)*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 A013609","(\u03b8+1)\u00b2"],"edges":["e-55a1228129c4451e","e-4729595bc722e8eb"],"text":"A086645 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A086645 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}