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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":"73c7d6f1d88183fe2e0550f74d08cea67f1154c06026b60550a4ea048dd6fb1f","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-5a70a6e97131d437","role":"sequence","name":"Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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,  6,   6;\n  1: 1, 10,  24,   16;\n  2: 1, 12,  39,   48,   20;\n  3: 1, 16,  75,  152,  140,    48;\n  4: 1, 18,  99,  252,  330,   216,    56;\n  5: 1, 22, 153,  516,  960,  1008,   560,   128;\n  6: 1, 24, 186,  720, 1605,  2160,  1736,   768,   144;\n  7: 1, 28, 258, 1216, 3405,  6012,  6776,  4736,  1872,   320;\n  8: 1, 30, 300, 1560, 4925, 10098, 13790, 12480,  7200,  2400,  352;\n  9: 1, 34, 390, 2360, 8825, 21918, 37352, 44000, 35280, 18400, 5632, 768;\n","recurrence_sentence":"The row polynomials satisfy (8+(18*n+54)*x+(9*n^2+81*n+122)*x^2+(24*n^2+112*n+108)*x^3+(16*n^2+48*n+32)*x^4)*P(n,x) = (8+(18*n+96)*x+(9*n^2+153*n+444)*x^2+(51*n^2+493*n+1024)*x^3+(106*n^2+758*n+1252)*x^4+(96*n^2+560*n+776)*x^5+(32*n^2+160*n+192)*x^6)*P(n-2,x) + (6*x+26*x^2+36*x^3+16*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 1 + 6*x + 6*x^2, P(1,x) = 1 + 10*x + 24*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1 + 6*x + 6*x^2;\nP[1, x_] := 1 + 10*x + 24*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + (18*n + 96)*x + (9*n^2 + 153*n + 444)*x^2 + (51*n^2 + 493*n + 1024)*x^3 + (106*n^2 + 758*n + 1252)*x^4 + (96*n^2 + 560*n + 776)*x^5 + (32*n^2 + 160*n + 192)*x^6)*P[n - 2, x] + (6*x + 26*x^2 + 36*x^3 + 16*x^4)*P[n - 1, x])/(8 + (18*n + 54)*x + (9*n^2 + 81*n + 122)*x^2 + (24*n^2 + 112*n + 108)*x^3 + (16*n^2 + 48*n + 32)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 6*x + 6*x^2; v[2] = 1 + 10*x + 24*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = ((8 + (18*n + 96)*x + (9*n^2 + 153*n + 444)*x^2 + (51*n^2 + 493*n + 1024)*x^3 + (106*n^2 + 758*n + 1252)*x^4 + (96*n^2 + 560*n + 776)*x^5 + (32*n^2 + 160*n + 192)*x^6)*v[i - 2] + (6*x + 26*x^2 + 36*x^3 + 16*x^4)*v[i - 1])/(8 + (18*n + 54)*x + (9*n^2 + 81*n + 122)*x^2 + (24*n^2 + 112*n + 108)*x^3 + (16*n^2 + 48*n + 32)*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-5a70a6e97131d437","steps":["\u03b8+1"],"edges":["e-cb26430ba269e72d"],"text":"A064861 \u2192 \u03b8+1"},"headline":"A064861 \u2192 \u03b8+1","chain":null}