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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":"cea121c58b161a0f9e21e77a8ea73d5f8fb136194ef2a53ee42c11f3928928e8","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-6a6e9e081117dc37","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.","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:   4,        176,            36;\n  3:   8,       2256,          5076,            128;\n  4:  16,      25744,        203040,         102976,             400;\n  5:  32,     285616,       5740596,       10205504,         1785100,            1152;\n  6:  64,    3147792,     139232088,      591919232,       386755800,        28330128,            3136;\n  7: 128,   34640048,    3117520764,    26571558784,     41518060600,     12470083056,       424340588,          8192;\n  8: 256,  381073808,   66715061328,  1031061384128,   3155372605600,   2319888114288,    363226445008,    6097180928,       20736;\n  9: 512, 4191887664, 1390033270980, 36542635134144, 195622729409700, 281696730349968, 111911820098316, 9884681038080, 84885725196, 51200;\n","recurrence_sentence":"The row polynomials satisfy (81*n^2-126*n+49)*P(n,x) = (1134*n^2-630*n-196-(5832*n^3-13608*n^2-15660*n+38080)*x+(59049*n^4-196830*n^3+199017*n^2-63504*n)*x^2)*P(n-2,x) - ((2916*n^2-1620*n-6741)*x-(5832*n^3+18468*n^2-6507*n-34370)*x^2-(26244*n^4-173502*n^3+224289*n^2-83412*n+1225)*x^3)*d/dx P(n-2,x) - ((6561*n^2-3645*n-6237)*x^2-(26244*n^3-34992*n^2+13365*n-6678)*x^3+(26244*n^3-41553*n^2+17010*n-441)*x^4)*d^2/dx^2 P(n-2,x) - (405*n^2-63*n-196-(2916*n^3-6804*n^2+5292*n-1372)*x)*P(n-1,x) + ((1458*n^2-1539*n-252)*x-(2916*n^3-3888*n^2+756*n+392)*x^2)*d/dx 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[((1134*n^2 - 630*n - 196 - (5832*n^3 - 13608*n^2 - 15660*n + 38080)*x + (59049*n^4 - 196830*n^3 + 199017*n^2 - 63504*n)*x^2)*P[n - 2, x] - ((2916*n^2 - 1620*n - 6741)*x - (5832*n^3 + 18468*n^2 - 6507*n - 34370)*x^2 - (26244*n^4 - 173502*n^3 + 224289*n^2 - 83412*n + 1225)*x^3)*D[P[n - 2, x], x] - ((6561*n^2 - 3645*n - 6237)*x^2 - (26244*n^3 - 34992*n^2 + 13365*n - 6678)*x^3 + (26244*n^3 - 41553*n^2 + 17010*n - 441)*x^4)*D[P[n - 2, x], {x, 2}] - (405*n^2 - 63*n - 196 - (2916*n^3 - 6804*n^2 + 5292*n - 1372)*x)*P[n - 1, x] + ((1458*n^2 - 1539*n - 252)*x - (2916*n^3 - 3888*n^2 + 756*n + 392)*x^2)*D[P[n - 1, x], x])/(81*n^2 - 126*n + 49)]];\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] = ((1134*n^2 - 630*n - 196 - (5832*n^3 - 13608*n^2 - 15660*n + 38080)*x + (59049*n^4 - 196830*n^3 + 199017*n^2 - 63504*n)*x^2)*v[i - 2] - ((2916*n^2 - 1620*n - 6741)*x - (5832*n^3 + 18468*n^2 - 6507*n - 34370)*x^2 - (26244*n^4 - 173502*n^3 + 224289*n^2 - 83412*n + 1225)*x^3)*deriv(v[i - 2], x) - ((6561*n^2 - 3645*n - 6237)*x^2 - (26244*n^3 - 34992*n^2 + 13365*n - 6678)*x^3 + (26244*n^3 - 41553*n^2 + 17010*n - 441)*x^4)*deriv(deriv(v[i - 2], x), x) - (405*n^2 - 63*n - 196 - (2916*n^3 - 6804*n^2 + 5292*n - 1372)*x)*v[i - 1] + ((1458*n^2 - 1539*n - 252)*x - (2916*n^3 - 3888*n^2 + 756*n + 392)*x^2)*deriv(v[i - 1], x))/(81*n^2 - 126*n + 49)); 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-6a6e9e081117dc37","steps":["(\u03b8+1)\u00b2"],"edges":["e-0c59ff9ad50c53fe"],"text":"A257619 \u2192 (\u03b8+1)\u00b2"},"headline":"A257619 \u2192 (\u03b8+1)\u00b2","chain":null}