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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":"e7351a1d3edad3881f7c539ceeb0ea01710f172773d07d13ed2293279774929c","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-0af3ffe51a3d7181","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 7.","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,         4;\n  2: 1,        64,           9;\n  3: 1,       572,        1287,            16;\n  4: 1,      4664,       38610,         18656,            25;\n  5: 1,     37428,      810018,       1440032,        233925,             36;\n  6: 1,    299568,    14592447,      63361408,      40534575,        2696112,            49;\n  7: 1,   2396716,   243151713,    2146272688,    3353551075,      972606852,      29359771,           64;\n  8: 1,  19173928,  3879157968,   62779708768,  194505962350,   141254344728,   21119860048,    306782848,         81;\n  9: 1, 153391652, 60344436420, 1677693668672, 9172031526350, 13207725397944, 5137936860308, 429115992320, 3106180953, 100;\n","recurrence_sentence":"The row polynomials satisfy (49*n^2-84*n+36)*P(n,x) = (294*n^2-210*n-36-(1372*n^3-3528*n^2-6580*n+15600)*x+(21609*n^4-78890*n^3+90797*n^2-36204*n+2340)*x^2)*P(n-2,x) - ((686*n^2-490*n-2436)*x-(1372*n^3+7448*n^2-3787*n-13380)*x^2-(9604*n^4-66542*n^3+97559*n^2-44772*n+3420)*x^3)*d/dx P(n-2,x) - ((2401*n^2-1715*n-2352)*x^2-(9604*n^3-15092*n^2+7105*n-3108)*x^3+(9604*n^3-17493*n^2+8820*n-756)*x^4)*d^2/dx^2 P(n-2,x) - (245*n^2-126*n-72-(1372*n^3-3528*n^2+3024*n-864)*x)*P(n-1,x) + ((686*n^2-833*n-84)*x-(1372*n^3-2156*n^2+672*n+144)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((294*n^2 - 210*n - 36 - (1372*n^3 - 3528*n^2 - 6580*n + 15600)*x + (21609*n^4 - 78890*n^3 + 90797*n^2 - 36204*n + 2340)*x^2)*P[n - 2, x] - ((686*n^2 - 490*n - 2436)*x - (1372*n^3 + 7448*n^2 - 3787*n - 13380)*x^2 - (9604*n^4 - 66542*n^3 + 97559*n^2 - 44772*n + 3420)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 1715*n - 2352)*x^2 - (9604*n^3 - 15092*n^2 + 7105*n - 3108)*x^3 + (9604*n^3 - 17493*n^2 + 8820*n - 756)*x^4)*D[P[n - 2, x], {x, 2}] - (245*n^2 - 126*n - 72 - (1372*n^3 - 3528*n^2 + 3024*n - 864)*x)*P[n - 1, x] + ((686*n^2 - 833*n - 84)*x - (1372*n^3 - 2156*n^2 + 672*n + 144)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 84*n + 36)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((294*n^2 - 210*n - 36 - (1372*n^3 - 3528*n^2 - 6580*n + 15600)*x + (21609*n^4 - 78890*n^3 + 90797*n^2 - 36204*n + 2340)*x^2)*v[i - 2] - ((686*n^2 - 490*n - 2436)*x - (1372*n^3 + 7448*n^2 - 3787*n - 13380)*x^2 - (9604*n^4 - 66542*n^3 + 97559*n^2 - 44772*n + 3420)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 1715*n - 2352)*x^2 - (9604*n^3 - 15092*n^2 + 7105*n - 3108)*x^3 + (9604*n^3 - 17493*n^2 + 8820*n - 756)*x^4)*deriv(deriv(v[i - 2], x), x) - (245*n^2 - 126*n - 72 - (1372*n^3 - 3528*n^2 + 3024*n - 864)*x)*v[i - 1] + ((686*n^2 - 833*n - 84)*x - (1372*n^3 - 2156*n^2 + 672*n + 144)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 84*n + 36)); 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-0af3ffe51a3d7181","steps":["(\u03b8+1)\u00b2"],"edges":["e-0c2d9cd733cace06"],"text":"A142462 \u2192 (\u03b8+1)\u00b2"},"headline":"A142462 \u2192 (\u03b8+1)\u00b2","chain":null}