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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":"ad22bfdb9a6949d90896a207f368e07cb0f5234dcc198f732a473768556a1753","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-118f91023ad73cd8","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,        4;\n   3:      13,       32,        9;\n   4:      63,      228,      135,       16;\n   5:     321,     1536,     1404,      384,       25;\n   6:    1683,    10020,    12420,     5440,      875,      36;\n   7:    8989,    64032,   100305,    60800,    16125,    1728,      49;\n   8:   48639,   403396,   764127,   593040,   221375,   40068,    3087,     64;\n   9:  265729,  2514944,  5589360,  5293568,  2558500,  661248,   87808,   5120,   81;\n  10: 1462563, 15554628, 39679632, 44389632, 26390700, 8917776, 1712256, 175104, 8019, 100;\n","recurrence_sentence":"The row polynomials satisfy (16*n-16+(40*n^2-82*n+42)*x+(36*n^3-116*n^2+125*n-45)*x^2+(12*n^4-50*n^3+76*n^2-50*n+12)*x^3+(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x^4)*P(n,x) = (112*n-176+(336*n^2-766*n+366)*x+(300*n^3-728*n^2+323*n+105)*x^2+(92*n^4-160*n^3-138*n^2+278*n-56)*x^3+(7*n^5+21*n^4-129*n^3+149*n^2-45*n+1)*x^4+(3*n^5-9*n^4+5*n^3+5*n^2-4*n)*x^5)*P(n-1,x) - (16+(96*n-72)*x+(120*n^2-28*n-96)*x^2+(48*n^3+100*n^2-182*n+12)*x^3+(4*n^4+68*n^3-64*n^2-40*n+20)*x^4+(6*n^4+18*n^3-46*n^2+20*n)*x^5+(2*n^4-2*n^3-2*n^2+2*n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((112*n - 176 + (336*n^2 - 766*n + 366)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (92*n^4 - 160*n^3 - 138*n^2 + 278*n - 56)*x^3 + (7*n^5 + 21*n^4 - 129*n^3 + 149*n^2 - 45*n + 1)*x^4 + (3*n^5 - 9*n^4 + 5*n^3 + 5*n^2 - 4*n)*x^5)*P[n - 1, x] - (16 + (96*n - 72)*x + (120*n^2 - 28*n - 96)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (4*n^4 + 68*n^3 - 64*n^2 - 40*n + 20)*x^4 + (6*n^4 + 18*n^3 - 46*n^2 + 20*n)*x^5 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^6)*D[P[n - 1, x], x])/(16*n - 16 + (40*n^2 - 82*n + 42)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (12*n^4 - 50*n^3 + 76*n^2 - 50*n + 12)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((112*n - 176 + (336*n^2 - 766*n + 366)*x + (300*n^3 - 728*n^2 + 323*n + 105)*x^2 + (92*n^4 - 160*n^3 - 138*n^2 + 278*n - 56)*x^3 + (7*n^5 + 21*n^4 - 129*n^3 + 149*n^2 - 45*n + 1)*x^4 + (3*n^5 - 9*n^4 + 5*n^3 + 5*n^2 - 4*n)*x^5)*v[i - 1] - (16 + (96*n - 72)*x + (120*n^2 - 28*n - 96)*x^2 + (48*n^3 + 100*n^2 - 182*n + 12)*x^3 + (4*n^4 + 68*n^3 - 64*n^2 - 40*n + 20)*x^4 + (6*n^4 + 18*n^3 - 46*n^2 + 20*n)*x^5 + (2*n^4 - 2*n^3 - 2*n^2 + 2*n)*x^6)*deriv(v[i - 1], x))/(16*n - 16 + (40*n^2 - 82*n + 42)*x + (36*n^3 - 116*n^2 + 125*n - 45)*x^2 + (12*n^4 - 50*n^3 + 76*n^2 - 50*n + 12)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-32321abfdad5764e","e-13b677a3dd9b539b"],"text":"A104684 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A104684 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}