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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":"a6d854ad5a2a01ed2311f603dc883787761af77d6fb6ac283bd358ed0411b608","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-64714e0ad28cd8ed","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 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,        240,           81;\n  3:    27,       3012,         6777,            432;\n  4:    81,      32712,       230418,         130848,           2025;\n  5:   243,     337164,      5683554,       10104096,        2107275,            8748;\n  6:   729,    3408576,    120137607,      484996608,      333715575,        30677184,          35721;\n  7:  2187,   34216980,   2333772567,    18260835600,    28532555625,      9335090268,      419158005,        139968;\n  8:  6561,  342624696,  43138602864,   595919356704,  1769018448750,   1340818552584,   234865726704,    5481995136,      531441;\n  9: 19683, 3427795356, 773443338120, 17753152790016, 90222283715550, 129920088550392, 54369030419424, 5500041515520, 69412855959, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (49*n^2-56*n+16)*P(n,x) = (588*n^2-84*n-144-(4116*n^3-7056*n^2-5572*n+10208)*x+(21609*n^4-54194*n^3+33761*n^2-1624*n-2288)*x^2)*P(n-2,x) - ((2058*n^2-294*n-2464)*x-(4116*n^3+6664*n^2+357*n-10008)*x^2-(9604*n^4-55566*n^3+45227*n^2-4424*n-2896)*x^3)*d/dx P(n-2,x) - ((2401*n^2-343*n-1960)*x^2-(9604*n^3-6860*n^2+833*n-1400)*x^3+(9604*n^3-9261*n^2+1176*n+560)*x^4)*d^2/dx^2 P(n-2,x) - (49*n^2+140*n-96-(1372*n^3-2352*n^2+1344*n-256)*x)*P(n-1,x) + ((686*n^2-441*n-168)*x-(1372*n^3-980*n^2-224*n+192)*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[((588*n^2 - 84*n - 144 - (4116*n^3 - 7056*n^2 - 5572*n + 10208)*x + (21609*n^4 - 54194*n^3 + 33761*n^2 - 1624*n - 2288)*x^2)*P[n - 2, x] - ((2058*n^2 - 294*n - 2464)*x - (4116*n^3 + 6664*n^2 + 357*n - 10008)*x^2 - (9604*n^4 - 55566*n^3 + 45227*n^2 - 4424*n - 2896)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 343*n - 1960)*x^2 - (9604*n^3 - 6860*n^2 + 833*n - 1400)*x^3 + (9604*n^3 - 9261*n^2 + 1176*n + 560)*x^4)*D[P[n - 2, x], {x, 2}] - (49*n^2 + 140*n - 96 - (1372*n^3 - 2352*n^2 + 1344*n - 256)*x)*P[n - 1, x] + ((686*n^2 - 441*n - 168)*x - (1372*n^3 - 980*n^2 - 224*n + 192)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 56*n + 16)]];\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] = ((588*n^2 - 84*n - 144 - (4116*n^3 - 7056*n^2 - 5572*n + 10208)*x + (21609*n^4 - 54194*n^3 + 33761*n^2 - 1624*n - 2288)*x^2)*v[i - 2] - ((2058*n^2 - 294*n - 2464)*x - (4116*n^3 + 6664*n^2 + 357*n - 10008)*x^2 - (9604*n^4 - 55566*n^3 + 45227*n^2 - 4424*n - 2896)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 343*n - 1960)*x^2 - (9604*n^3 - 6860*n^2 + 833*n - 1400)*x^3 + (9604*n^3 - 9261*n^2 + 1176*n + 560)*x^4)*deriv(deriv(v[i - 2], x), x) - (49*n^2 + 140*n - 96 - (1372*n^3 - 2352*n^2 + 1344*n - 256)*x)*v[i - 1] + ((686*n^2 - 441*n - 168)*x - (1372*n^3 - 980*n^2 - 224*n + 192)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 56*n + 16)); 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-64714e0ad28cd8ed","steps":["(\u03b8+1)\u00b2"],"edges":["e-a08065070bea6d6d"],"text":"A257627 \u2192 (\u03b8+1)\u00b2"},"headline":"A257627 \u2192 (\u03b8+1)\u00b2","chain":null}