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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":"3518d0c77e9ef0ddceaacabaa284158754771cb10ce3b5d10eb24f01fc48abfe","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-3e465972400fd143","role":"sequence","name":"shift_x_plus_one of Triangular array of Motzkin polynomial coefficients.","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:   1;\n  2:   2,    4;\n  3:   4,   12;\n  4:   9,   40,    18;\n  5:  21,  120,    90;\n  6:  51,  360,   405,    80;\n  7: 127, 1064,  1575,   560;\n  8: 323, 3136,  5796,  3136,  350;\n  9: 835, 9216, 20412, 14784, 3150;\n","recurrence_sentence":"The row polynomials satisfy (36*n+72+(60*n^2+168*n+96)*x+(46*n^3+122*n^2+72*n+24)*x^2+(12*n^4+45*n^3+35*n^2-26*n-24)*x^3+(n^5+4*n^4+5*n^3+2*n^2)*x^4)*P(n,x) = (180*n-72+(426*n^2+318*n-420)*x+(302*n^3+826*n^2-120*n-744)*x^2+(72*n^4+493*n^3+411*n^2-478*n-456)*x^3+(5*n^5+88*n^4+209*n^3+14*n^2-240*n-64)*x^4+(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16)*x^5)*P(n-1,x) - (54+(216*n+324)*x+(222*n^2+954*n+750)*x^2+(72*n^3+788*n^2+1644*n+852)*x^3+(6*n^4+204*n^3+1004*n^2+1386*n+508)*x^4+(14*n^4+180*n^3+542*n^2+576*n+168)*x^5+(8*n^4+48*n^3+104*n^2+96*n+32)*x^6)*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[((180*n - 72 + (426*n^2 + 318*n - 420)*x + (302*n^3 + 826*n^2 - 120*n - 744)*x^2 + (72*n^4 + 493*n^3 + 411*n^2 - 478*n - 456)*x^3 + (5*n^5 + 88*n^4 + 209*n^3 + 14*n^2 - 240*n - 64)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*P[n - 1, x] - (54 + (216*n + 324)*x + (222*n^2 + 954*n + 750)*x^2 + (72*n^3 + 788*n^2 + 1644*n + 852)*x^3 + (6*n^4 + 204*n^3 + 1004*n^2 + 1386*n + 508)*x^4 + (14*n^4 + 180*n^3 + 542*n^2 + 576*n + 168)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*D[P[n - 1, x], x])/(36*n + 72 + (60*n^2 + 168*n + 96)*x + (46*n^3 + 122*n^2 + 72*n + 24)*x^2 + (12*n^4 + 45*n^3 + 35*n^2 - 26*n - 24)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^4)]];\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] = ((180*n - 72 + (426*n^2 + 318*n - 420)*x + (302*n^3 + 826*n^2 - 120*n - 744)*x^2 + (72*n^4 + 493*n^3 + 411*n^2 - 478*n - 456)*x^3 + (5*n^5 + 88*n^4 + 209*n^3 + 14*n^2 - 240*n - 64)*x^4 + (4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16)*x^5)*v[i - 1] - (54 + (216*n + 324)*x + (222*n^2 + 954*n + 750)*x^2 + (72*n^3 + 788*n^2 + 1644*n + 852)*x^3 + (6*n^4 + 204*n^3 + 1004*n^2 + 1386*n + 508)*x^4 + (14*n^4 + 180*n^3 + 542*n^2 + 576*n + 168)*x^5 + (8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x^6)*deriv(v[i - 1], x))/(36*n + 72 + (60*n^2 + 168*n + 96)*x + (46*n^3 + 122*n^2 + 72*n + 24)*x^2 + (12*n^4 + 45*n^3 + 35*n^2 - 26*n - 24)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^4)); 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-169bd483754ab390","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-5fd319f73e442e4d","e-0aba537821e4147d"],"text":"A055151 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A055151 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}