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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":"1aec946bf99e4f74d73b779a6c09d63ca295cd03b8e8d2f453eceabe75dec83f","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-54b58a0ddb0e34fe","role":"sequence","name":"A decomposition of the double factorials A001147.","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;\n  2: 1,     8;\n  3: 1,    40,      36;\n  4: 1,   144,     540,      128;\n  5: 1,   464,    4644,     4736,       400;\n  6: 1,  1432,   31572,    82688,     33200,      1152;\n  7: 1,  4344,  190080,  1027840,   1074000,    203904,      3136;\n  8: 1, 13088, 1067688, 10574080,  22514000,  11522304,   1150912,    8192;\n  9: 1, 39328, 5750712, 96795904, 367846000, 386106624, 109060672, 6135808, 20736;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2-n-(8*n^3-24*n^2+8*n+16)*x+(36*n^4-152*n^3+212*n^2-112*n+16)*x^2)*P(n-2,x) - ((4*n^2-4*n-3)*x-(8*n^3-4*n-14)*x^2-(16*n^4-120*n^3+212*n^2-128*n+20)*x^3)*d/dx P(n-2,x) - ((4*n^2-4*n-2)*x^2-(16*n^3-32*n^2+20*n-6)*x^3+(16*n^3-36*n^2+24*n-4)*x^4)*d^2/dx^2 P(n-2,x) - (n-1-(8*n^3-24*n^2+24*n-8)*x)*P(n-1,x) + ((4*n^2-6*n+1)*x-(8*n^3-16*n^2+8*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - n - (8*n^3 - 24*n^2 + 8*n + 16)*x + (36*n^4 - 152*n^3 + 212*n^2 - 112*n + 16)*x^2)*P[n - 2, x] - ((4*n^2 - 4*n - 3)*x - (8*n^3 - 4*n - 14)*x^2 - (16*n^4 - 120*n^3 + 212*n^2 - 128*n + 20)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4*n - 2)*x^2 - (16*n^3 - 32*n^2 + 20*n - 6)*x^3 + (16*n^3 - 36*n^2 + 24*n - 4)*x^4)*D[P[n - 2, x], {x, 2}] - (n - 1 - (8*n^3 - 24*n^2 + 24*n - 8)*x)*P[n - 1, x] + ((4*n^2 - 6*n + 1)*x - (8*n^3 - 16*n^2 + 8*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - n - (8*n^3 - 24*n^2 + 8*n + 16)*x + (36*n^4 - 152*n^3 + 212*n^2 - 112*n + 16)*x^2)*v[i - 2] - ((4*n^2 - 4*n - 3)*x - (8*n^3 - 4*n - 14)*x^2 - (16*n^4 - 120*n^3 + 212*n^2 - 128*n + 20)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4*n - 2)*x^2 - (16*n^3 - 32*n^2 + 20*n - 6)*x^3 + (16*n^3 - 36*n^2 + 24*n - 4)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - 1 - (8*n^3 - 24*n^2 + 24*n - 8)*x)*v[i - 1] + ((4*n^2 - 6*n + 1)*x - (8*n^3 - 16*n^2 + 8*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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-54b58a0ddb0e34fe","steps":["(\u03b8+1)\u00b2"],"edges":["e-9b2a4a437176effe"],"text":"A185410 \u2192 (\u03b8+1)\u00b2"},"headline":"A185410 \u2192 (\u03b8+1)\u00b2","chain":null}