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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":"f5203eb099b410215fb7a64a6836cda5e008875748d32cad3ec639f2e92ba960","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,           168,              9;\n  1:         4536,          4452,            567,             16;\n  2:        54432,         92736,          20979,           1344,            25;\n  3:       598752,       1649088,         586845,          63840,          2625,           36;\n  4:      6158592,      26127360,       13662432,        2237760,        152250,         4536,          49;\n  5:     60046272,     378473472,      278215560,       64172304,       6410250,       310716,        7203,         64;\n  6:    560431872,    5099687424,     5106258144,     1588833792,     218098125,     15268176,      569037,      10752,        81;\n  7:   5043886848,   64704407040,    86163670944,    35090025984,    6344772525,    600677532,    31959711,     962304,     15309,     100;\n  8:  44019376128,  780121165824,  1355659224192,   706462248960,  163353294000,  20055075768,  1426777443,   60802560,   1530900,   21000,   121;\n  9: 374164697088, 9000739657728, 20095876906560, 13166442971136, 3808678887000, 588908652228, 53732961009, 3036039552, 107469180, 2320500, 27951, 144;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(36*n^2+288*n+252+(48*n+336)*x-22*x^2)*P(n-2,x) - ((432*n+3024)*x-(24*n+24)*x^2-26*x^3)*d/dx P(n-2,x) - (1296*x^2-144*x^3-5*x^4)*d^2/dx^2 P(n-2,x) + (12*n^2+78*n+108+(8*n+16)*x)*P(n-1,x) + ((72*n+144)*x-(4*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 336 + 168*x + 9*x^2, P(1,x) = 4536 + 4452*x + 567*x^2 + 16*x^3.","mathematica":"P[0, x_] := 336 + 168*x + 9*x^2;\nP[1, x_] := 4536 + 4452*x + 567*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(36*n^2 + 288*n + 252 + (48*n + 336)*x - 22*x^2)*P[n - 2, x] - ((432*n + 3024)*x - (24*n + 24)*x^2 - 26*x^3)*D[P[n - 2, x], x] - (1296*x^2 - 144*x^3 - 5*x^4)*D[P[n - 2, x], {x, 2}] + (12*n^2 + 78*n + 108 + (8*n + 16)*x)*P[n - 1, x] + ((72*n + 144)*x - (4*n + 8)*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 + 168*x + 9*x^2; v[2] = 4536 + 4452*x + 567*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n^2 + 288*n + 252 + (48*n + 336)*x - 22*x^2)*v[i - 2] - ((432*n + 3024)*x - (24*n + 24)*x^2 - 26*x^3)*deriv(v[i - 2], x) - (1296*x^2 - 144*x^3 - 5*x^4)*deriv(deriv(v[i - 2], x), x) + (12*n^2 + 78*n + 108 + (8*n + 16)*x)*v[i - 1] + ((72*n + 144)*x - (4*n + 8)*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)\u00b2"],"edges":["e-a3c3c10c52830971"],"text":"A132166 \u2192 (\u03b8+1)\u00b2"},"headline":"A132166 \u2192 (\u03b8+1)\u00b2","chain":null}