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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":"7c9d037afd86fe3d81ce6018a4b6975bae08c3ad252ff5e97fd92f3e3ff65624","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-b9dfb234567937e9","role":"sequence","name":"basis_map_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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,     28,       9;\n  3:    33,    140,      99,      16;\n  4:   109,    620,     684,     240,      25;\n  5:   360,   2568,    3861,    2128,     475,     36;\n  6:  1189,  10200,   19440,   14704,    5150,    828,     49;\n  7:  3927,  39364,   90909,   87664,   42225,  10620,   1323,    64;\n  8: 12970, 148756,  403686,  473872,  291775, 100908,  19600,  1984,   81;\n  9: 42837, 553220, 1725237, 2388560, 1794950, 794304, 211925, 33344, 2835, 100;\n","recurrence_sentence":"The row polynomials satisfy (169+(507*n-104)*x+(442*n^2-78*n-66)*x^2+(117*n^3+62*n^2-117*n-4)*x^3+(9*n^4+12*n^3-31*n^2-6*n+5)*x^4-(3*n^3+2*n^2)*x^5)*P(n,x) = (169+(507*n+572)*x+(442*n^2+1313*n+701)*x^2+(117*n^3+855*n^2+1164*n+356)*x^3+(9*n^4+165*n^3+472*n^2+362*n+47)*x^4+(9*n^4+45*n^3+48*n^2-9*n-16)*x^5-(3*n^3+11*n^2+13*n+5)*x^6)*P(n-2,x) + (507+(1521*n+364)*x+(1326*n^2+1430*n-250)*x^2+(351*n^3+1096*n^2+303*n-384)*x^3+(27*n^4+207*n^3+215*n^2-127*n-123)*x^4+(9*n^4+21*n^3-25*n^2-48*n-16)*x^5-(3*n^3+8*n^2+7*n+2)*x^6)*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[((169 + (507*n + 572)*x + (442*n^2 + 1313*n + 701)*x^2 + (117*n^3 + 855*n^2 + 1164*n + 356)*x^3 + (9*n^4 + 165*n^3 + 472*n^2 + 362*n + 47)*x^4 + (9*n^4 + 45*n^3 + 48*n^2 - 9*n - 16)*x^5 - (3*n^3 + 11*n^2 + 13*n + 5)*x^6)*P[n - 2, x] + (507 + (1521*n + 364)*x + (1326*n^2 + 1430*n - 250)*x^2 + (351*n^3 + 1096*n^2 + 303*n - 384)*x^3 + (27*n^4 + 207*n^3 + 215*n^2 - 127*n - 123)*x^4 + (9*n^4 + 21*n^3 - 25*n^2 - 48*n - 16)*x^5 - (3*n^3 + 8*n^2 + 7*n + 2)*x^6)*P[n - 1, x])/(169 + (507*n - 104)*x + (442*n^2 - 78*n - 66)*x^2 + (117*n^3 + 62*n^2 - 117*n - 4)*x^3 + (9*n^4 + 12*n^3 - 31*n^2 - 6*n + 5)*x^4 - (3*n^3 + 2*n^2)*x^5)]];\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] = ((169 + (507*n + 572)*x + (442*n^2 + 1313*n + 701)*x^2 + (117*n^3 + 855*n^2 + 1164*n + 356)*x^3 + (9*n^4 + 165*n^3 + 472*n^2 + 362*n + 47)*x^4 + (9*n^4 + 45*n^3 + 48*n^2 - 9*n - 16)*x^5 - (3*n^3 + 11*n^2 + 13*n + 5)*x^6)*v[i - 2] + (507 + (1521*n + 364)*x + (1326*n^2 + 1430*n - 250)*x^2 + (351*n^3 + 1096*n^2 + 303*n - 384)*x^3 + (27*n^4 + 207*n^3 + 215*n^2 - 127*n - 123)*x^4 + (9*n^4 + 21*n^3 - 25*n^2 - 48*n - 16)*x^5 - (3*n^3 + 8*n^2 + 7*n + 2)*x^6)*v[i - 1])/(169 + (507*n - 104)*x + (442*n^2 - 78*n - 66)*x^2 + (117*n^3 + 62*n^2 - 117*n - 4)*x^3 + (9*n^4 + 12*n^3 - 31*n^2 - 6*n + 5)*x^4 - (3*n^3 + 2*n^2)*x^5)); 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-d5c14b9fec875ecc","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-03e3acc3469b904b","e-66e4d070ca64dd6a"],"text":"A121574 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A121574 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}