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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":"8aa2b1dfbda8ce4f6cb4b4f6d9ae6268868c9fd4725a035c41630f1e21b277e3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-141ab74a8d7e58b7","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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:      5,      16;\n   3:     21,     120,       90;\n   4:     84,     672,      945,      320;\n   5:    330,    3360,     6804,     4480,      875;\n   6:   1287,   15840,    41580,    40320,    15750,     2016;\n   7:   5005,   72072,   231660,   295680,   173250,    45360,     4116;\n   8:  19448,  320320,  1216215,  1921920,  1501500,   598752,   113190,    7680;\n   9:  75582, 1400256,  6126120, 11531520, 11261250,  6054048,  1765764,  253440,  13365;\n  10: 293930, 6046560, 29930472, 65345280, 76576500, 51891840, 20600580, 4612608, 521235, 22000;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+8*n^3-3*n^2-7*n-2+(6*n^5+12*n^4-18*n^3-33*n^2+15*n+18)*x+(n^6+3*n^5-3*n^4-11*n^3+6*n^2+12*n-8)*x^2)*P(n,x) = (16*n^4+24*n^3+12*n^2+2*n+(24*n^5+84*n^4+28*n^3+18*n+8)*x+(4*n^6+40*n^5+69*n^4-10*n^3+58*n^2+23*n-22)*x^2+(3*n^6+15*n^5+14*n^4+3*n^3+36*n^2-23*n+6)*x^3)*P(n-1,x) - ((8*n^3+12*n^2+6*n+1)*x+(12*n^4+44*n^3+20*n^2+2*n+3)*x^2+(2*n^5+23*n^4+48*n^3+3*n^2+6*n-1)*x^3+(2*n^5+11*n^4+12*n^3-5*n^2+10*n-3)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^4 + 24*n^3 + 12*n^2 + 2*n + (24*n^5 + 84*n^4 + 28*n^3 + 18*n + 8)*x + (4*n^6 + 40*n^5 + 69*n^4 - 10*n^3 + 58*n^2 + 23*n - 22)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*P[n - 1, x] - ((8*n^3 + 12*n^2 + 6*n + 1)*x + (12*n^4 + 44*n^3 + 20*n^2 + 2*n + 3)*x^2 + (2*n^5 + 23*n^4 + 48*n^3 + 3*n^2 + 6*n - 1)*x^3 + (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 8*n^3 - 3*n^2 - 7*n - 2 + (6*n^5 + 12*n^4 - 18*n^3 - 33*n^2 + 15*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^4 + 24*n^3 + 12*n^2 + 2*n + (24*n^5 + 84*n^4 + 28*n^3 + 18*n + 8)*x + (4*n^6 + 40*n^5 + 69*n^4 - 10*n^3 + 58*n^2 + 23*n - 22)*x^2 + (3*n^6 + 15*n^5 + 14*n^4 + 3*n^3 + 36*n^2 - 23*n + 6)*x^3)*v[i - 1] - ((8*n^3 + 12*n^2 + 6*n + 1)*x + (12*n^4 + 44*n^3 + 20*n^2 + 2*n + 3)*x^2 + (2*n^5 + 23*n^4 + 48*n^3 + 3*n^2 + 6*n - 1)*x^3 + (2*n^5 + 11*n^4 + 12*n^3 - 5*n^2 + 10*n - 3)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 8*n^3 - 3*n^2 - 7*n - 2 + (6*n^5 + 12*n^4 - 18*n^3 - 33*n^2 + 15*n + 18)*x + (n^6 + 3*n^5 - 3*n^4 - 11*n^3 + 6*n^2 + 12*n - 8)*x^2)); 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-a2304b34e998691c","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-1067be4ea4032d33","e-91d082935a189390"],"text":"A062145 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A062145 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}