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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":"fc83d1c7b5c0c5cdd3a02ada72075de24b1c89c06e01e2901e87a665653effc0","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-0515531ee2138839","role":"sequence","name":"reversal of Riordan array (1, 3+3x).","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:     3;\n  2:     9,     12;\n  3:    27,     72;\n  4:    81,    324,     81;\n  5:   243,   1296,    729;\n  6:   729,   4860,   4374,    432;\n  7:  2187,  17496,  21870,   5184;\n  8:  6561,  61236,  98415,  38880,  2025;\n  9: 19683, 209952, 413343, 233280, 30375;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2+9*n+(9*n^3-6*n)*x+(n^4+n^3)*x^2)*P(n,x) = (27*n^2+27*n+(27*n^3+72*n^2-18*n-72)*x+(3*n^4+33*n^3+24*n^2-42*n-12)*x^2+(2*n^4+8*n^3+6*n^2-8*n-8)*x^3)*P(n-1,x) - ((27*n+27)*x+(27*n^2+90*n+54)*x^2+(3*n^3+51*n^2+96*n+36)*x^3+(4*n^3+20*n^2+32*n+16)*x^4)*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[((27*n^2 + 27*n + (27*n^3 + 72*n^2 - 18*n - 72)*x + (3*n^4 + 33*n^3 + 24*n^2 - 42*n - 12)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*P[n - 1, x] - ((27*n + 27)*x + (27*n^2 + 90*n + 54)*x^2 + (3*n^3 + 51*n^2 + 96*n + 36)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*D[P[n - 1, x], x])/(9*n^2 + 9*n + (9*n^3 - 6*n)*x + (n^4 + n^3)*x^2)]];\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] = ((27*n^2 + 27*n + (27*n^3 + 72*n^2 - 18*n - 72)*x + (3*n^4 + 33*n^3 + 24*n^2 - 42*n - 12)*x^2 + (2*n^4 + 8*n^3 + 6*n^2 - 8*n - 8)*x^3)*v[i - 1] - ((27*n + 27)*x + (27*n^2 + 90*n + 54)*x^2 + (3*n^3 + 51*n^2 + 96*n + 36)*x^3 + (4*n^3 + 20*n^2 + 32*n + 16)*x^4)*deriv(v[i - 1], x))/(9*n^2 + 9*n + (9*n^3 - 6*n)*x + (n^4 + n^3)*x^2)); 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-016d7724d51f6416","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-450f652a77090025","e-06b0e846638b7591"],"text":"A099093 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A099093 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}