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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":"6a382fc3eee2473413a94c360e18fa4530675af5ecb03dd850f2eee23b5994ee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3bc26d3108ee6bfd","role":"sequence","name":"Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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: 1,        4;\n  2: 1,       56,           9;\n  3: 1,      444,         999,           16;\n  4: 1,     3184,       25974,        12736,            25;\n  5: 1,    22388,      473778,       842272,        139925,            36;\n  6: 1,   156840,     7418439,     32006336,      20606775,       1411560,            49;\n  7: 1,  1098028,   107379297,    937828784,    1465357475,     429517188,      13450843,           64;\n  8: 1,  7686368,  1487341692,  23736539264,   73267873750,   53407213344,    8097749212,    122981888,         81;\n  9: 1, 53804772, 20079098100, 548828277312, 2981435464350, 4293267068664, 1680786599268, 142784697600, 1089546633, 100;\n","recurrence_sentence":"The row polynomials satisfy (36*n^2-60*n+25)*P(n,x) = (180*n^2-120*n-25-(864*n^3-2160*n^2-3384*n+8140)*x+(11664*n^4-41472*n^3+46008*n^2-17280*n+825)*x^2)*P(n-2,x) - ((432*n^2-288*n-1320)*x-(864*n^3+3888*n^2-1800*n-7100)*x^2-(5184*n^4-35424*n^3+50112*n^2-21720*n+1300)*x^3)*d/dx P(n-2,x) - ((1296*n^2-864*n-1260)*x^2-(5184*n^3-7776*n^2+3456*n-1560)*x^3+(5184*n^3-9072*n^2+4320*n-300)*x^4)*d^2/dx^2 P(n-2,x) - (144*n^2-60*n-50-(864*n^3-2160*n^2+1800*n-500)*x)*P(n-1,x) + ((432*n^2-504*n-60)*x-(864*n^3-1296*n^2+360*n+100)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((180*n^2 - 120*n - 25 - (864*n^3 - 2160*n^2 - 3384*n + 8140)*x + (11664*n^4 - 41472*n^3 + 46008*n^2 - 17280*n + 825)*x^2)*P[n - 2, x] - ((432*n^2 - 288*n - 1320)*x - (864*n^3 + 3888*n^2 - 1800*n - 7100)*x^2 - (5184*n^4 - 35424*n^3 + 50112*n^2 - 21720*n + 1300)*x^3)*D[P[n - 2, x], x] - ((1296*n^2 - 864*n - 1260)*x^2 - (5184*n^3 - 7776*n^2 + 3456*n - 1560)*x^3 + (5184*n^3 - 9072*n^2 + 4320*n - 300)*x^4)*D[P[n - 2, x], {x, 2}] - (144*n^2 - 60*n - 50 - (864*n^3 - 2160*n^2 + 1800*n - 500)*x)*P[n - 1, x] + ((432*n^2 - 504*n - 60)*x - (864*n^3 - 1296*n^2 + 360*n + 100)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 60*n + 25)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((180*n^2 - 120*n - 25 - (864*n^3 - 2160*n^2 - 3384*n + 8140)*x + (11664*n^4 - 41472*n^3 + 46008*n^2 - 17280*n + 825)*x^2)*v[i - 2] - ((432*n^2 - 288*n - 1320)*x - (864*n^3 + 3888*n^2 - 1800*n - 7100)*x^2 - (5184*n^4 - 35424*n^3 + 50112*n^2 - 21720*n + 1300)*x^3)*deriv(v[i - 2], x) - ((1296*n^2 - 864*n - 1260)*x^2 - (5184*n^3 - 7776*n^2 + 3456*n - 1560)*x^3 + (5184*n^3 - 9072*n^2 + 4320*n - 300)*x^4)*deriv(deriv(v[i - 2], x), x) - (144*n^2 - 60*n - 50 - (864*n^3 - 2160*n^2 + 1800*n - 500)*x)*v[i - 1] + ((432*n^2 - 504*n - 60)*x - (864*n^3 - 1296*n^2 + 360*n + 100)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 60*n + 25)); 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-3bc26d3108ee6bfd","steps":["(\u03b8+1)\u00b2"],"edges":["e-f0ce31309acbe067"],"text":"A142461 \u2192 (\u03b8+1)\u00b2"},"headline":"A142461 \u2192 (\u03b8+1)\u00b2","chain":null}