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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":"ec6d061d8001fdb850959269ac7453dd80f9242650e8414ee5d7038fb7ebc0b8","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-efe494309672b404","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:   0;\n  1:   1;\n  2:  10,     4;\n  3:  35,    84,      9;\n  4:  84,   504,    324,      16;\n  5: 165,  1848,   2970,     880,      25;\n  6: 286,  5148,  15444,   11440,    1950,     36;\n  7: 455, 12012,  57915,   80080,   34125,   3780,     49;\n  8: 680, 24752, 175032,  388960,  309400,  85680,   6664,    64;\n  9: 969, 46512, 453492, 1478048, 1889550, 976752, 189924, 10944, 81;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-5*n+3+(4*n^4-20*n^3+37*n^2-30*n+9)*x)*P(n,x) = (2*n^2+n+(4*n^4-8*n^3+13*n^2-23*n+8)*x+(12*n^4-40*n^3+37*n^2-10*n+4)*x^2)*P(n-1,x) + ((4*n-2)*x+(8*n^3-12*n^2)*x^2-(8*n^3-12*n^2+4*n-2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (4*n^4 - 8*n^3 + 13*n^2 - 23*n + 8)*x + (12*n^4 - 40*n^3 + 37*n^2 - 10*n + 4)*x^2)*P[n - 1, x] + ((4*n - 2)*x + (8*n^3 - 12*n^2)*x^2 - (8*n^3 - 12*n^2 + 4*n - 2)*x^3)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3 + (4*n^4 - 20*n^3 + 37*n^2 - 30*n + 9)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + n + (4*n^4 - 8*n^3 + 13*n^2 - 23*n + 8)*x + (12*n^4 - 40*n^3 + 37*n^2 - 10*n + 4)*x^2)*v[i - 1] + ((4*n - 2)*x + (8*n^3 - 12*n^2)*x^2 - (8*n^3 - 12*n^2 + 4*n - 2)*x^3)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 3 + (4*n^4 - 20*n^3 + 37*n^2 - 30*n + 9)*x)); 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-c30cb06ad75c5e21","steps":["D","\u03b8+1"],"edges":["e-1287908e85a1078d","e-7d3ba7cd8506b9b0"],"text":"A103327 \u2192 D \u2192 \u03b8+1"},"headline":"A103327 \u2192 D \u2192 \u03b8+1","chain":null}