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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":"63ac5a871cf72ebeb65dfc6c8b523ea1dbb8b394517e014bbae4ff1d4bdac6a4","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-8159f24e9c9fb552","role":"sequence","name":"shift_x_plus_one of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph 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;\n  1:    1;\n  2:    2,     4;\n  3:    4,    12;\n  4:   10,    48,    27;\n  5:   26,   160,   135;\n  6:   76,   600,   810,   240;\n  7:  232,  2184,  3780,  1680;\n  8:  764,  8512, 18900, 13440,  2625;\n  9: 2620, 33408, 88452, 80640, 23625;\n","recurrence_sentence":"The row polynomials satisfy (64+(80*n+96)*x+(56*n^2+40*n+32)*x^2+(12*n^3+28*n^2-8*n-16)*x^3+(n^4+2*n^3+n^2)*x^4)*P(n,x) = (64*n+(136*n^2+184*n-144)*x+(80*n^3+304*n^2+96*n-352)*x^2+(16*n^4+144*n^3+176*n^2-96*n-224)*x^3+(n^5+22*n^4+65*n^3+12*n^2-80*n-16)*x^4+(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^5)*P(n-1,x) - (32+(96*n+192)*x+(88*n^2+392*n+416)*x^2+(24*n^3+280*n^2+624*n+416)*x^3+(2*n^4+60*n^3+322*n^2+480*n+200)*x^4+(4*n^4+48*n^3+156*n^2+176*n+48)*x^5+(2*n^4+12*n^3+26*n^2+24*n+8)*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[((64*n + (136*n^2 + 184*n - 144)*x + (80*n^3 + 304*n^2 + 96*n - 352)*x^2 + (16*n^4 + 144*n^3 + 176*n^2 - 96*n - 224)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*P[n - 1, x] - (32 + (96*n + 192)*x + (88*n^2 + 392*n + 416)*x^2 + (24*n^3 + 280*n^2 + 624*n + 416)*x^3 + (2*n^4 + 60*n^3 + 322*n^2 + 480*n + 200)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*D[P[n - 1, x], x])/(64 + (80*n + 96)*x + (56*n^2 + 40*n + 32)*x^2 + (12*n^3 + 28*n^2 - 8*n - 16)*x^3 + (n^4 + 2*n^3 + n^2)*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] = ((64*n + (136*n^2 + 184*n - 144)*x + (80*n^3 + 304*n^2 + 96*n - 352)*x^2 + (16*n^4 + 144*n^3 + 176*n^2 - 96*n - 224)*x^3 + (n^5 + 22*n^4 + 65*n^3 + 12*n^2 - 80*n - 16)*x^4 + (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^5)*v[i - 1] - (32 + (96*n + 192)*x + (88*n^2 + 392*n + 416)*x^2 + (24*n^3 + 280*n^2 + 624*n + 416)*x^3 + (2*n^4 + 60*n^3 + 322*n^2 + 480*n + 200)*x^4 + (4*n^4 + 48*n^3 + 156*n^2 + 176*n + 48)*x^5 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^6)*deriv(v[i - 1], x))/(64 + (80*n + 96)*x + (56*n^2 + 40*n + 32)*x^2 + (12*n^3 + 28*n^2 - 8*n - 16)*x^3 + (n^4 + 2*n^3 + n^2)*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-97b51734d39ede6c","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-d700e7dd8b41ed69","e-9c24c73a245587e6"],"text":"A100861 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A100861 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}