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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":"b77f60765cacf01607ba39d05224672238f1e0654c18ddc30b8b98e2a1caf90d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-44b1dbf5d009bf35","role":"sequence","name":"A convolution triangle of numbers obtained from A036224.","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:          336,            84,             3;\n  1:         4536,          2226,           189,             4;\n  2:        54432,         46368,          6993,           336,            5;\n  3:       598752,        824544,        195615,         15960,          525,           6;\n  4:      6158592,      13063680,       4554144,        559440,        30450,         756,          7;\n  5:     60046272,     189236736,      92738520,      16043076,      1282050,       51786,       1029,         8;\n  6:    560431872,    2549843712,    1702086048,     397208448,     43619625,     2544696,      81291,      1344,        9;\n  7:   5043886848,   32352203520,   28721223648,    8772506496,   1268954505,   100112922,    4565673,    120288,     1701,     10;\n  8:  44019376128,  390060582912,  451886408064,  176615562240,  32670658800,  3342512628,  203825349,   7600320,   170100,   2100,   11;\n  9: 374164697088, 4500369828864, 6698625635520, 3291610742784, 761735777400, 98151442038, 7676137287, 379504944, 11941020, 232050, 2541, 12;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(36*n^2+288*n+252+(24*n+24)*x+2*x^2)*P(n-2,x) - ((216*n+216)*x+(6*n+42)*x^2+x^3)*d/dx P(n-2,x) + (12*n^2+78*n+108+(4*n+8)*x)*P(n-1,x) + ((36*n+72)*x+(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 336 + 84*x + 3*x^2, P(1,x) = 4536 + 2226*x + 189*x^2 + 4*x^3.","mathematica":"P[0, x_] := 336 + 84*x + 3*x^2;\nP[1, x_] := 4536 + 2226*x + 189*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(36*n^2 + 288*n + 252 + (24*n + 24)*x + 2*x^2)*P[n - 2, x] - ((216*n + 216)*x + (6*n + 42)*x^2 + x^3)*D[P[n - 2, x], x] + (12*n^2 + 78*n + 108 + (4*n + 8)*x)*P[n - 1, x] + ((36*n + 72)*x + (n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 336 + 84*x + 3*x^2; v[2] = 4536 + 2226*x + 189*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n^2 + 288*n + 252 + (24*n + 24)*x + 2*x^2)*v[i - 2] - ((216*n + 216)*x + (6*n + 42)*x^2 + x^3)*deriv(v[i - 2], x) + (12*n^2 + 78*n + 108 + (4*n + 8)*x)*v[i - 1] + ((36*n + 72)*x + (n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n + 6)); 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-44b1dbf5d009bf35","steps":["\u03b8+1"],"edges":["e-d720a3737b3683f1"],"text":"A132166 \u2192 \u03b8+1"},"headline":"A132166 \u2192 \u03b8+1","chain":null}