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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":"3d883e243f524d31f4ddcb79068b13d0b24ecfb994183de61c47170168b8ff71","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-a6f3b35778192ffb","role":"sequence","name":"Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).","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,  2;\n  2: 1,  6,    3;\n  3: 1, 12,   18,    4;\n  4: 1, 20,   57,   40,     5;\n  5: 1, 30,  135,  180,    75,     6;\n  6: 1, 42,  270,  564,   450,   126,     7;\n  7: 1, 56,  483, 1428,  1785,   966,   196,    8;\n  8: 1, 72,  798, 3136,  5535,  4704,  1862,  288,   9;\n  9: 1, 90, 1242, 6216, 14535, 17442, 10878, 3312, 405, 10;\n","recurrence_sentence":"The row polynomials satisfy -(6*n-6-(n^3-6*n^2+17*n-12)*x+(5*n^3-18*n^2+13*n)*x^2-(4*n^3-6*n^2+2*n)*x^3)*P(n,x) = -(6*n-6-(n^3-6*n^2+17*n-12)*x+(5*n^3-22*n^2-37*n+54)*x^2-(n^3+28*n^2-17*n-12)*x^3-(12*n^3-2*n^2-16*n+6)*x^4)*P(n-1,x) - (12*x-(2*n^2-10*n-16)*x^2+(6*n^2+18*n+12)*x^3+(6*n^2+18*n)*x^4+(8*n^2+8*n-4)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 - (n^3 - 6*n^2 + 17*n - 12)*x + (5*n^3 - 22*n^2 - 37*n + 54)*x^2 - (n^3 + 28*n^2 - 17*n - 12)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*P[n - 1, x] - (12*x - (2*n^2 - 10*n - 16)*x^2 + (6*n^2 + 18*n + 12)*x^3 + (6*n^2 + 18*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*D[P[n - 1, x], x])/(-6*n + 6 + (n^3 - 6*n^2 + 17*n - 12)*x - (5*n^3 - 18*n^2 + 13*n)*x^2 + (4*n^3 - 6*n^2 + 2*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 6 - (n^3 - 6*n^2 + 17*n - 12)*x + (5*n^3 - 22*n^2 - 37*n + 54)*x^2 - (n^3 + 28*n^2 - 17*n - 12)*x^3 - (12*n^3 - 2*n^2 - 16*n + 6)*x^4)*v[i - 1] - (12*x - (2*n^2 - 10*n - 16)*x^2 + (6*n^2 + 18*n + 12)*x^3 + (6*n^2 + 18*n)*x^4 + (8*n^2 + 8*n - 4)*x^5)*deriv(v[i - 1], x))/(-6*n + 6 + (n^3 - 6*n^2 + 17*n - 12)*x - (5*n^3 - 18*n^2 + 13*n)*x^2 + (4*n^3 - 6*n^2 + 2*n)*x^3)); 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-a6f3b35778192ffb","steps":["\u03b8+1"],"edges":["e-bb559b3f89099a24"],"text":"A056241 \u2192 \u03b8+1"},"headline":"A056241 \u2192 \u03b8+1","chain":null}