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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":"3ccd37890c817f6602ad3fd49286b54df6bc350d60faaa9c70e5a2505e4a2427","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c9dce8c25621fdea","role":"sequence","name":"Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  6,    3;\n   4: 1, 12,   24,     4;\n   5: 1, 20,   87,    96,      5;\n   6: 1, 30,  225,   580,    445,      6;\n   7: 1, 42,  480,  2180,   4070,   2490,       7;\n   8: 1, 56,  903,  6300,  20895,  31458,   16604,       8;\n   9: 1, 72, 1554, 15344,  78295, 209160,  270326,  128576,       9;\n  10: 1, 90, 2502, 33096, 238875, 979146, 2232482, 2573520, 1131057, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n-4)*P(n,x) = -((6*n^2-24*n+18)*x^2+(2*n^3-24*n^2+82*n-84)*x^3)*P(n-3,x) - ((2*n^2-8*n+6)*x^3+(n^3-10*n^2+31*n-30)*x^4)*d/dx P(n-3,x) - (2*n-2-(4*n-8)*x+(6*n^2-24*n+24)*x^2)*P(n-2,x) + ((2*n-2)*x+(3*n^2-13*n+14)*x^2+(n^3-9*n^2+24*n-20)*x^3)*d/dx P(n-2,x) + (4*n-6+(4*n^2-16*n+16)*x)*P(n-1,x) - ((2*n-2)*x+(n^2-7*n+10)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 1 + 6*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[3, x_] := 1 + 6*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-((6*n^2 - 24*n + 18)*x^2 + (2*n^3 - 24*n^2 + 82*n - 84)*x^3)*P[n - 3, x] - ((2*n^2 - 8*n + 6)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*D[P[n - 3, x], x] - (2*n - 2 - (4*n - 8)*x + (6*n^2 - 24*n + 24)*x^2)*P[n - 2, x] + ((2*n - 2)*x + (3*n^2 - 13*n + 14)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*D[P[n - 2, x], x] + (4*n - 6 + (4*n^2 - 16*n + 16)*x)*P[n - 1, x] - ((2*n - 2)*x + (n^2 - 7*n + 10)*x^2)*D[P[n - 1, x], x])/(2*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 6*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-((6*n^2 - 24*n + 18)*x^2 + (2*n^3 - 24*n^2 + 82*n - 84)*x^3)*v[i - 3] - ((2*n^2 - 8*n + 6)*x^3 + (n^3 - 10*n^2 + 31*n - 30)*x^4)*deriv(v[i - 3], x) - (2*n - 2 - (4*n - 8)*x + (6*n^2 - 24*n + 24)*x^2)*v[i - 2] + ((2*n - 2)*x + (3*n^2 - 13*n + 14)*x^2 + (n^3 - 9*n^2 + 24*n - 20)*x^3)*deriv(v[i - 2], x) + (4*n - 6 + (4*n^2 - 16*n + 16)*x)*v[i - 1] - ((2*n - 2)*x + (n^2 - 7*n + 10)*x^2)*deriv(v[i - 1], x))/(2*n - 4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["\u03b8+1"],"edges":["e-e1869a5dbf71f6ee"],"text":"A046716 \u2192 \u03b8+1"},"headline":"A046716 \u2192 \u03b8+1","chain":null}