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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":"39fab4d4d0c1ab573723b64126ba218d4d4918d833682717fdc149b2c70b343a","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-5aa52e3acf29a627","role":"sequence","name":"Triangle read by rows, related to clusters of type D.","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:      4,      12;\n  2:     15,      80,       54;\n  3:     56,     420,      540,      160;\n  4:    210,    2016,     3780,     2240,      375;\n  5:    792,    9240,    22680,    20160,     7000,      756;\n  6:   3003,   41184,   124740,   147840,    78750,    18144,    1372;\n  7:  11440,  180180,   648648,   960960,   693000,   249480,   41160,    2304;\n  8:  43758,  777920,  3243240,  5765760,  5255250,  2594592,  679140,   84480,   3645;\n  9: 167960, 3325608, 15752880, 32672640, 36036000, 22702680, 8240232, 1647360, 160380, 5500;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4+16*n^3+20*n^2+8*n+(6*n^5+27*n^4+36*n^3+8*n^2-8*n)*x+(n^6+6*n^5+12*n^4+8*n^3)*x^2)*P(n,x) = (16*n^4+56*n^3+72*n^2+40*n+8+(24*n^5+144*n^4+286*n^3+258*n^2+116*n+24)*x+(4*n^6+52*n^5+194*n^4+298*n^3+244*n^2+120*n+24)*x^2+(3*n^6+24*n^5+69*n^4+96*n^3+84*n^2+40*n+8)*x^3)*P(n-1,x) - ((8*n^3+24*n^2+24*n+8)*x+(12*n^4+68*n^3+119*n^2+86*n+24)*x^2+(2*n^5+28*n^4+104*n^3+147*n^2+94*n+24)*x^3+(2*n^5+16*n^4+44*n^3+52*n^2+32*n+8)*x^4)*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[((16*n^4 + 56*n^3 + 72*n^2 + 40*n + 8 + (24*n^5 + 144*n^4 + 286*n^3 + 258*n^2 + 116*n + 24)*x + (4*n^6 + 52*n^5 + 194*n^4 + 298*n^3 + 244*n^2 + 120*n + 24)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*P[n - 1, x] - ((8*n^3 + 24*n^2 + 24*n + 8)*x + (12*n^4 + 68*n^3 + 119*n^2 + 86*n + 24)*x^2 + (2*n^5 + 28*n^4 + 104*n^3 + 147*n^2 + 94*n + 24)*x^3 + (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 16*n^3 + 20*n^2 + 8*n + (6*n^5 + 27*n^4 + 36*n^3 + 8*n^2 - 8*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*n^3)*x^2)]];\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] = ((16*n^4 + 56*n^3 + 72*n^2 + 40*n + 8 + (24*n^5 + 144*n^4 + 286*n^3 + 258*n^2 + 116*n + 24)*x + (4*n^6 + 52*n^5 + 194*n^4 + 298*n^3 + 244*n^2 + 120*n + 24)*x^2 + (3*n^6 + 24*n^5 + 69*n^4 + 96*n^3 + 84*n^2 + 40*n + 8)*x^3)*v[i - 1] - ((8*n^3 + 24*n^2 + 24*n + 8)*x + (12*n^4 + 68*n^3 + 119*n^2 + 86*n + 24)*x^2 + (2*n^5 + 28*n^4 + 104*n^3 + 147*n^2 + 94*n + 24)*x^3 + (2*n^5 + 16*n^4 + 44*n^3 + 52*n^2 + 32*n + 8)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 16*n^3 + 20*n^2 + 8*n + (6*n^5 + 27*n^4 + 36*n^3 + 8*n^2 - 8*n)*x + (n^6 + 6*n^5 + 12*n^4 + 8*n^3)*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-5aa52e3acf29a627","steps":["(\u03b8+1)\u00b2"],"edges":["e-e9e569a876b76c11"],"text":"A345013 \u2192 (\u03b8+1)\u00b2"},"headline":"A345013 \u2192 (\u03b8+1)\u00b2","chain":null}