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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":"546eb6d9ff6f83f24a47fe9eec599084fc66ff6caf42700d4b3f0d69fe39c59a","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-286e213a8791bd45","role":"sequence","name":"T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.","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;\n  2:  3,   8;\n  3:  4,  32;\n  4:  5,  80,   36;\n  5:  6, 160,  216;\n  6:  7, 280,  756,   128;\n  7:  8, 448, 2016,  1024;\n  8:  9, 672, 4536,  4608,  400;\n  9: 10, 960, 9072, 15360, 4000;\n","recurrence_sentence":"The row polynomials satisfy (n-(5*n^3+n^2-2*n)*x+(n^5+2*n^4+n^3)*x^2)*P(n,x) = (n+1-(5*n^3+6*n^2-9*n+6)*x+(n^5+3*n^4-15*n^3-19*n^2+26*n+12)*x^2+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^3)*P(n-1,x) + (2*x-(10*n^2+22*n+12)*x^2+(2*n^4+12*n^3+46*n^2+68*n+24)*x^3-(4*n^4+24*n^3+52*n^2+48*n+16)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 1 - (5*n^3 + 6*n^2 - 9*n + 6)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*P[n - 1, x] + (2*x - (10*n^2 + 22*n + 12)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*D[P[n - 1, x], x])/(n - (5*n^3 + n^2 - 2*n)*x + (n^5 + 2*n^4 + n^3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 - (5*n^3 + 6*n^2 - 9*n + 6)*x + (n^5 + 3*n^4 - 15*n^3 - 19*n^2 + 26*n + 12)*x^2 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^3)*v[i - 1] + (2*x - (10*n^2 + 22*n + 12)*x^2 + (2*n^4 + 12*n^3 + 46*n^2 + 68*n + 24)*x^3 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^4)*deriv(v[i - 1], x))/(n - (5*n^3 + n^2 - 2*n)*x + (n^5 + 2*n^4 + n^3)*x^2)); 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-286e213a8791bd45","steps":["(\u03b8+1)\u00b2"],"edges":["e-dd4db257447b93f9"],"text":"A105070 \u2192 (\u03b8+1)\u00b2"},"headline":"A105070 \u2192 (\u03b8+1)\u00b2","chain":null}