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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":"ff0fe81f567b216aadadb4619bb2bd5b042586b2db0576ba81295e9efffbc18a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8eb9bb3a09f0b7fd","role":"sequence","name":"Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=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,  24,    36;\n  1: 1,  48,   180,    128;\n  2: 1,  80,   540,    896,     400;\n  3: 1, 120,  1260,   3584,    3600,    1152;\n  4: 1, 168,  2520,  10752,   18000,   12672,     3136;\n  5: 1, 224,  4536,  26880,   66000,   76032,    40768,     8192;\n  6: 1, 288,  7560,  59136,  198000,  329472,   285376,   122880,    20736;\n  7: 1, 360, 11880, 118272,  514800, 1153152,  1426880,   983040,   352512,   51200;\n  8: 1, 440, 17820, 219648, 1201200, 3459456,  5707520,  5570560,  3172608,  972800,  123904;\n  9: 1, 528, 25740, 384384, 2574000, 9225216, 19405568, 25067520, 20093184, 9728000, 2601984, 294912;\n","recurrence_sentence":"The row polynomials satisfy (18*n+27-(22*n^3+99*n^2+149*n+75)*x+(2*n^5+15*n^4+44*n^3+63*n^2+44*n+12)*x^2)*P(n,x) = (18*n+27-(22*n^3+99*n^2+77*n-69)*x+(2*n^5+15*n^4+6*n^3-165*n^2-398*n-264)*x^2+(2*n^5+20*n^4+76*n^3+136*n^2+114*n+36)*x^3)*P(n-1,x) + (36*x-(44*n^2+176*n+138)*x^2+(4*n^4+32*n^3+66*n^2+8*n-42)*x^3+(2*n^4+16*n^3+44*n^2+48*n+18)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 24*x + 36*x^2.","mathematica":"P[0, x_] := 1 + 24*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n + 27 - (22*n^3 + 99*n^2 + 77*n - 69)*x + (2*n^5 + 15*n^4 + 6*n^3 - 165*n^2 - 398*n - 264)*x^2 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^3)*P[n - 1, x] + (36*x - (44*n^2 + 176*n + 138)*x^2 + (4*n^4 + 32*n^3 + 66*n^2 + 8*n - 42)*x^3 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^4)*D[P[n - 1, x], x])/(18*n + 27 - (22*n^3 + 99*n^2 + 149*n + 75)*x + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 24*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18*n + 27 - (22*n^3 + 99*n^2 + 77*n - 69)*x + (2*n^5 + 15*n^4 + 6*n^3 - 165*n^2 - 398*n - 264)*x^2 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^3)*v[i - 1] + (36*x - (44*n^2 + 176*n + 138)*x^2 + (4*n^4 + 32*n^3 + 66*n^2 + 8*n - 42)*x^3 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^4)*deriv(v[i - 1], x))/(18*n + 27 - (22*n^3 + 99*n^2 + 149*n + 75)*x + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*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-8eb9bb3a09f0b7fd","steps":["(\u03b8+1)\u00b2"],"edges":["e-ac14b3745c4c6962"],"text":"A123519 \u2192 (\u03b8+1)\u00b2"},"headline":"A123519 \u2192 (\u03b8+1)\u00b2","chain":null}