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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":"12429cb3f111bcc049626f2d32f3201df624f61b4c11820435ff6a541876930c","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-930be05ae3b321d9","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.","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:     3,        12;\n  2:     9,       168,          81;\n  3:    27,      1572,        3537,          432;\n  4:    81,     12624,       77814,        50496,          2025;\n  5:   243,     94524,     1282014,      2279136,        590775,          8748;\n  6:   729,    684024,    18143055,     68374080,      50397375,       6156216,         35721;\n  7:  2187,   4866900,   234971847,   1638443280,    2560067625,     939887388,      59619525,       139968;\n  8:  6561,  34339488,  2880354492,  34184386944,   97282569750,   76914870624,   15681930012,    549431808,     531441;\n  9: 19683, 241294956, 34079078700, 651022819776, 3076870231050, 4430693132712, 1993757385564, 242340115200, 4886222859, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (16*n^2-8*n+1)*P(n,x) = (48*n^2+24*n-9-(768*n^3-576*n^2-880*n+500)*x+(2304*n^4-2816*n^3-544*n^2+656*n-95)*x^2)*P(n-2,x) - ((384*n^2+192*n-184)*x-(768*n^3+1216*n^2+528*n-540)*x^2-(1024*n^4-4608*n^3-256*n^2+928*n-148)*x^3)*d/dx P(n-2,x) - ((256*n^2+128*n-112)*x^2-(1024*n^3+256*n^2-64*n-80)*x^3+(1024*n^3-192*n+32)*x^4)*d^2/dx^2 P(n-2,x) + (32*n^2-32*n+6+(256*n^3-192*n^2+48*n-4)*x)*P(n-1,x) + ((128*n^2-24)*x-(256*n^3+64*n^2-80*n+12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[((48*n^2 + 24*n - 9 - (768*n^3 - 576*n^2 - 880*n + 500)*x + (2304*n^4 - 2816*n^3 - 544*n^2 + 656*n - 95)*x^2)*P[n - 2, x] - ((384*n^2 + 192*n - 184)*x - (768*n^3 + 1216*n^2 + 528*n - 540)*x^2 - (1024*n^4 - 4608*n^3 - 256*n^2 + 928*n - 148)*x^3)*D[P[n - 2, x], x] - ((256*n^2 + 128*n - 112)*x^2 - (1024*n^3 + 256*n^2 - 64*n - 80)*x^3 + (1024*n^3 - 192*n + 32)*x^4)*D[P[n - 2, x], {x, 2}] + (32*n^2 - 32*n + 6 + (256*n^3 - 192*n^2 + 48*n - 4)*x)*P[n - 1, x] + ((128*n^2 - 24)*x - (256*n^3 + 64*n^2 - 80*n + 12)*x^2)*D[P[n - 1, x], x])/(16*n^2 - 8*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = ((48*n^2 + 24*n - 9 - (768*n^3 - 576*n^2 - 880*n + 500)*x + (2304*n^4 - 2816*n^3 - 544*n^2 + 656*n - 95)*x^2)*v[i - 2] - ((384*n^2 + 192*n - 184)*x - (768*n^3 + 1216*n^2 + 528*n - 540)*x^2 - (1024*n^4 - 4608*n^3 - 256*n^2 + 928*n - 148)*x^3)*deriv(v[i - 2], x) - ((256*n^2 + 128*n - 112)*x^2 - (1024*n^3 + 256*n^2 - 64*n - 80)*x^3 + (1024*n^3 - 192*n + 32)*x^4)*deriv(deriv(v[i - 2], x), x) + (32*n^2 - 32*n + 6 + (256*n^3 - 192*n^2 + 48*n - 4)*x)*v[i - 1] + ((128*n^2 - 24)*x - (256*n^3 + 64*n^2 - 80*n + 12)*x^2)*deriv(v[i - 1], x))/(16*n^2 - 8*n + 1)); 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-930be05ae3b321d9","steps":["(\u03b8+1)\u00b2"],"edges":["e-fcd5197b3b450d75"],"text":"A257621 \u2192 (\u03b8+1)\u00b2"},"headline":"A257621 \u2192 (\u03b8+1)\u00b2","chain":null}