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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":"61f3d84768828321c1afdee2294c2f689e541a3f1e45e963b9c1e7a11b46ee28","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-1fe6eede339bc6ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (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:    3;\n  2:    8,     4;\n  3:   21,    24;\n  4:   55,   100,     9;\n  5:  144,   360,    81;\n  6:  377,  1200,   459,   16;\n  7:  987,  3816,  2106,  192;\n  8: 2584, 11756,  8559, 1376,  25;\n  9: 6765, 35400, 32157, 7680, 375;\n","recurrence_sentence":"The row polynomials satisfy (300*n-(375*n^2+195*n)*x-(75*n^3-180*n^2-147*n)*x^2+(45*n^4+15*n^3-36*n^2-18*n)*x^3+(3*n^5+6*n^4+3*n^3)*x^4)*P(n,x) = (700*n+200-(700*n^2+430*n+580)*x-(250*n^3+440*n^2-898*n-548)*x^2+(95*n^4+135*n^3-14*n^2-34*n-164)*x^3+(7*n^5+66*n^4+65*n^3-68*n^2-38*n+4)*x^4+(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^5)*P(n-1,x) + (125-(375*n+225)*x-(50*n^2+15)*x^2+(75*n^3+185*n^2+405*n+161)*x^3+(5*n^4+15*n^3+70*n^2+102*n-18)*x^4-(n^4+66*n^3+153*n^2+84*n+12)*x^5-(4*n^4+24*n^3+52*n^2+48*n+16)*x^6)*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[((700*n + 200 - (700*n^2 + 430*n + 580)*x - (250*n^3 + 440*n^2 - 898*n - 548)*x^2 + (95*n^4 + 135*n^3 - 14*n^2 - 34*n - 164)*x^3 + (7*n^5 + 66*n^4 + 65*n^3 - 68*n^2 - 38*n + 4)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*P[n - 1, x] + (125 - (375*n + 225)*x - (50*n^2 + 15)*x^2 + (75*n^3 + 185*n^2 + 405*n + 161)*x^3 + (5*n^4 + 15*n^3 + 70*n^2 + 102*n - 18)*x^4 - (n^4 + 66*n^3 + 153*n^2 + 84*n + 12)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*D[P[n - 1, x], x])/(300*n - (375*n^2 + 195*n)*x - (75*n^3 - 180*n^2 - 147*n)*x^2 + (45*n^4 + 15*n^3 - 36*n^2 - 18*n)*x^3 + (3*n^5 + 6*n^4 + 3*n^3)*x^4)]];\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] = ((700*n + 200 - (700*n^2 + 430*n + 580)*x - (250*n^3 + 440*n^2 - 898*n - 548)*x^2 + (95*n^4 + 135*n^3 - 14*n^2 - 34*n - 164)*x^3 + (7*n^5 + 66*n^4 + 65*n^3 - 68*n^2 - 38*n + 4)*x^4 + (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^5)*v[i - 1] + (125 - (375*n + 225)*x - (50*n^2 + 15)*x^2 + (75*n^3 + 185*n^2 + 405*n + 161)*x^3 + (5*n^4 + 15*n^3 + 70*n^2 + 102*n - 18)*x^4 - (n^4 + 66*n^3 + 153*n^2 + 84*n + 12)*x^5 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^6)*deriv(v[i - 1], x))/(300*n - (375*n^2 + 195*n)*x - (75*n^3 - 180*n^2 - 147*n)*x^2 + (45*n^4 + 15*n^3 - 36*n^2 - 18*n)*x^3 + (3*n^5 + 6*n^4 + 3*n^3)*x^4)); 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-1fe6eede339bc6ea","steps":["(\u03b8+1)\u00b2"],"edges":["e-56e9b74884f9da57"],"text":"A181371 \u2192 (\u03b8+1)\u00b2"},"headline":"A181371 \u2192 (\u03b8+1)\u00b2","chain":null}