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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":"d5e1b8e24e22ed63001e5355e7a64e0161501ebfd264d92ca088964553fa0360","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-f339d17ab0e2d2a9","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial 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:      3,       4;\n  2:     10,      26,       12;\n  3:     37,     142,      129,       32;\n  4:    151,     766,     1038,      516,       80;\n  5:    674,    4258,     7698,     5796,     1855,      192;\n  6:   3263,   24720,    56073,    56976,    28180,     6294,      448;\n  7:  17007,  150484,   411729,   529336,   358055,   126666,    20685,    1024;\n  8:  94828,  961026,  3085176,  4814372,  4158355,  2041950,   543466,   66824,   2304;\n  9: 562595, 6432226, 23737746, 43673668, 46092695, 29187138, 10958612, 2267560, 214011, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(x-(n+1)*x^3)*P(n,x) = (2*n+2+(n^2+7*n+3)*x-4*x^2-(n^3+8*n^2+16*n+6)*x^3-(2*n^3+10*n^2+12*n)*x^4-(n^3+3*n^2+2*n)*x^5)*P(n-2,x) - (1+(n+5)*x+(2*n+6)*x^2-(n^2+4*n+5)*x^3-(3*n^2+13*n+15)*x^4-(3*n^2+11*n+10)*x^5-(n^2+3*n+2)*x^6)*d/dx P(n-2,x) - (2+(2*n+5)*x-2*x^2-(2*n^2+8*n+8)*x^3-(2*n^2+4*n+2)*x^4)*P(n-1,x) + (x+2*x^2-n*x^3-(2*n+2)*x^4-(n+1)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (n^2 + 7*n + 3)*x - 4*x^2 - (n^3 + 8*n^2 + 16*n + 6)*x^3 - (2*n^3 + 10*n^2 + 12*n)*x^4 - (n^3 + 3*n^2 + 2*n)*x^5)*P[n - 2, x] - (1 + (n + 5)*x + (2*n + 6)*x^2 - (n^2 + 4*n + 5)*x^3 - (3*n^2 + 13*n + 15)*x^4 - (3*n^2 + 11*n + 10)*x^5 - (n^2 + 3*n + 2)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 5)*x - 2*x^2 - (2*n^2 + 8*n + 8)*x^3 - (2*n^2 + 4*n + 2)*x^4)*P[n - 1, x] + (x + 2*x^2 - n*x^3 - (2*n + 2)*x^4 - (n + 1)*x^5)*D[P[n - 1, x], x])/(-x + (n + 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 2 + (n^2 + 7*n + 3)*x - 4*x^2 - (n^3 + 8*n^2 + 16*n + 6)*x^3 - (2*n^3 + 10*n^2 + 12*n)*x^4 - (n^3 + 3*n^2 + 2*n)*x^5)*v[i - 2] - (1 + (n + 5)*x + (2*n + 6)*x^2 - (n^2 + 4*n + 5)*x^3 - (3*n^2 + 13*n + 15)*x^4 - (3*n^2 + 11*n + 10)*x^5 - (n^2 + 3*n + 2)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 5)*x - 2*x^2 - (2*n^2 + 8*n + 8)*x^3 - (2*n^2 + 4*n + 2)*x^4)*v[i - 1] + (x + 2*x^2 - n*x^3 - (2*n + 2)*x^4 - (n + 1)*x^5)*deriv(v[i - 1], x))/(-x + (n + 1)*x^3)); 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-0990b5be9f344750","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-2b15edbdb8f7710e","e-d993dead2d2067d6"],"text":"A126351 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A126351 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}