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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":"3896a582e486e1a772bb1cd10fc5fee7dc2be5bfd4cb91bd6dc734a0ac9f69ce","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-b5ebaf3676d81658","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/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:   1;\n  2:   3,    6;\n  3:   5,   12;\n  4:  11,   42,   27;\n  5:  21,   96,   81;\n  6:  43,  246,  324,  108;\n  7:  85,  564,  918,  432;\n  8: 171, 1314, 2673, 1944,  405;\n  9: 341, 2952, 7047, 6480, 2025;\n","recurrence_sentence":"The row polynomials satisfy (12*n+(9*n^2+13*n)*x+(3*n^3+3*n^2)*x^2)*P(n,x) = (60*n-48+(63*n^2+119*n-160)*x+(15*n^3+111*n^2+36*n-156)*x^2+(18*n^3+36*n^2-18*n-36)*x^3)*P(n-1,x) - (36+(54*n+180)*x+(18*n^2+207*n+329)*x^2+(51*n^2+261*n+258)*x^3+(36*n^2+108*n+72)*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[((60*n - 48 + (63*n^2 + 119*n - 160)*x + (15*n^3 + 111*n^2 + 36*n - 156)*x^2 + (18*n^3 + 36*n^2 - 18*n - 36)*x^3)*P[n - 1, x] - (36 + (54*n + 180)*x + (18*n^2 + 207*n + 329)*x^2 + (51*n^2 + 261*n + 258)*x^3 + (36*n^2 + 108*n + 72)*x^4)*D[P[n - 1, x], x])/(12*n + (9*n^2 + 13*n)*x + (3*n^3 + 3*n^2)*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] = ((60*n - 48 + (63*n^2 + 119*n - 160)*x + (15*n^3 + 111*n^2 + 36*n - 156)*x^2 + (18*n^3 + 36*n^2 - 18*n - 36)*x^3)*v[i - 1] - (36 + (54*n + 180)*x + (18*n^2 + 207*n + 329)*x^2 + (51*n^2 + 261*n + 258)*x^3 + (36*n^2 + 108*n + 72)*x^4)*deriv(v[i - 1], x))/(12*n + (9*n^2 + 13*n)*x + (3*n^3 + 3*n^2)*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-7d332fcb7bf9a0f7","steps":["basis A013610","\u03b8+1"],"edges":["e-95533c5935436804","e-cbbea85008780b3c"],"text":"A128100 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A128100 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}