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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":"ed08c6018e00887c1206af48e7909637d4df6583b04670eec1b8e70f17fd3479","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-966c4f25f43e716e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+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;\n  1:      4,       6;\n  2:     16,      72,       27;\n  3:     64,     480,      648,       108;\n  4:    256,    2688,     6480,      4320,       405;\n  5:   1024,   13824,    48384,     60480,     24300,      1458;\n  6:   4096,   67584,   311040,    580608,    453600,    122472,      5103;\n  7:  16384,  319488,  1824768,   4561920,   5443200,   2939328,    571536,    17496;\n  8:  65536, 1474560, 10063872,  31629312,  51321600,  43110144,  17146080,  2519424,    59049;\n  9: 262144, 6684672, 53084160, 201277440, 415134720, 480370176, 301771008, 92378880, 10628820, 196830;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-4*n+(12*n^3-12*n^2+3*n)*x)*P(n,x) = (32*n^2-16*n+(48*n^3+96*n^2-108*n)*x+(108*n^3-117*n+27)*x^2)*P(n-1,x) - (16*n*x+(24*n^2+72*n-6)*x^2+(72*n^2+72*n-18)*x^3)*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[((32*n^2 - 16*n + (48*n^3 + 96*n^2 - 108*n)*x + (108*n^3 - 117*n + 27)*x^2)*P[n - 1, x] - (16*n*x + (24*n^2 + 72*n - 6)*x^2 + (72*n^2 + 72*n - 18)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 4*n + (12*n^3 - 12*n^2 + 3*n)*x)]];\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] = ((32*n^2 - 16*n + (48*n^3 + 96*n^2 - 108*n)*x + (108*n^3 - 117*n + 27)*x^2)*v[i - 1] - (16*n*x + (24*n^2 + 72*n - 6)*x^2 + (72*n^2 + 72*n - 18)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 4*n + (12*n^3 - 12*n^2 + 3*n)*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":{"root":"n-c30cb06ad75c5e21","steps":["basis A013610","\u03b8+1"],"edges":["e-71859d6a54438fd0","e-5a4464deefd1005e"],"text":"A103327 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A103327 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}