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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":"0b798266c9696fcdc43c674aa1a6cd7d48501ce333e39e4c591c5f0ee0e9a7d8","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-c1d69b4cd7e1b71e","role":"sequence","name":"basis_map_A013609 of Pascal-(1,3,1) array.","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:     2,      8;\n  2:     7,     56,      36;\n  3:    20,    240,     432,      128;\n  4:    61,    976,    2808,     2176,      400;\n  5:   182,   3640,   14544,    19328,     8800,     1152;\n  6:   547,  13128,   67068,   126848,    99600,    31104,     3136;\n  7:  1640,  45920,  285984,   705280,   803200,   428544,   100352,    8192;\n  8:  4921, 157472, 1157040,  3523328,  5298400,  4124160,  1630720,  303104,  20736;\n  9: 14762, 531432, 4500576, 16326912, 30628800, 31732992, 18289152, 5677056, 870912, 51200;\n","recurrence_sentence":"The row polynomials satisfy (4+(24*n-4)*x+(44*n^2-4*n-6)*x^2+(24*n^3+20*n^2-16*n-5)*x^3+(4*n^4+8*n^3-6*n^2-4*n+2)*x^4-(2*n^3+2*n^2)*x^5)*P(n,x) = (12+(72*n+84)*x+(132*n^2+396*n+222)*x^2+(72*n^3+540*n^2+792*n+273)*x^3+(12*n^4+216*n^3+678*n^2+648*n+150)*x^4+(24*n^4+138*n^3+228*n^2+114*n+12)*x^5-(12*n^3+48*n^2+60*n+24)*x^6)*P(n-2,x) + (8+(48*n+24)*x+(88*n^2+128*n+12)*x^2+(48*n^3+200*n^2+120*n-50)*x^3+(8*n^4+80*n^3+140*n^2+12*n-48)*x^4+(8*n^4+28*n^3+10*n^2-16*n-6)*x^5-(4*n^3+12*n^2+12*n+4)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((12 + (72*n + 84)*x + (132*n^2 + 396*n + 222)*x^2 + (72*n^3 + 540*n^2 + 792*n + 273)*x^3 + (12*n^4 + 216*n^3 + 678*n^2 + 648*n + 150)*x^4 + (24*n^4 + 138*n^3 + 228*n^2 + 114*n + 12)*x^5 - (12*n^3 + 48*n^2 + 60*n + 24)*x^6)*P[n - 2, x] + (8 + (48*n + 24)*x + (88*n^2 + 128*n + 12)*x^2 + (48*n^3 + 200*n^2 + 120*n - 50)*x^3 + (8*n^4 + 80*n^3 + 140*n^2 + 12*n - 48)*x^4 + (8*n^4 + 28*n^3 + 10*n^2 - 16*n - 6)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*P[n - 1, x])/(4 + (24*n - 4)*x + (44*n^2 - 4*n - 6)*x^2 + (24*n^3 + 20*n^2 - 16*n - 5)*x^3 + (4*n^4 + 8*n^3 - 6*n^2 - 4*n + 2)*x^4 - (2*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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((12 + (72*n + 84)*x + (132*n^2 + 396*n + 222)*x^2 + (72*n^3 + 540*n^2 + 792*n + 273)*x^3 + (12*n^4 + 216*n^3 + 678*n^2 + 648*n + 150)*x^4 + (24*n^4 + 138*n^3 + 228*n^2 + 114*n + 12)*x^5 - (12*n^3 + 48*n^2 + 60*n + 24)*x^6)*v[i - 2] + (8 + (48*n + 24)*x + (88*n^2 + 128*n + 12)*x^2 + (48*n^3 + 200*n^2 + 120*n - 50)*x^3 + (8*n^4 + 80*n^3 + 140*n^2 + 12*n - 48)*x^4 + (8*n^4 + 28*n^3 + 10*n^2 - 16*n - 6)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*v[i - 1])/(4 + (24*n - 4)*x + (44*n^2 - 4*n - 6)*x^2 + (24*n^3 + 20*n^2 - 16*n - 5)*x^3 + (4*n^4 + 8*n^3 - 6*n^2 - 4*n + 2)*x^4 - (2*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-d8f9ad9ce0e039d0","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-471a582f60d8a81d","e-1489b6d0566d06f7"],"text":"A081578 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A081578 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}