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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":"d6e18e40513965c3babf6a7918dca9f172acee89de511541caf8b486c14bf876","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-eed4f744ce063213","role":"sequence","name":"Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^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,  12,   18;\n  2: 1,  24,  117,   192,    100;\n  3: 1,  36,  297,  1008,   1650,    1296,     392;\n  4: 1,  48,  558,  2880,   8025,   12960,   12152,     6144,     1296;\n  5: 1,  60,  900,  6240,  24625,   60588,   96530,    99840,    64800,     24000,      3872;\n  6: 1,  72, 1323, 11520,  58875,  195048,  440461,   693504,   763020,    576000,    284592,     82944,     10816;\n  7: 1,  84, 1827, 19152, 120225,  501228, 1467697,  3112896,  4852386,   5569200,   4655112,   2757888,   1097824,    263424,     28800;\n  8: 1,  96, 2412, 29568, 220150, 1106784, 3985660, 10646016, 21524049,  33268800,  39368560,  35417088,  23811424,  11590656,   3859200,   786432,    73984;\n  9: 1, 108, 3078, 43200, 372150, 2190888, 9368016, 30198528, 75122721, 146256300, 224440722, 271786752, 258480768, 190850688, 107179200, 44236800, 12651264, 2239488, 184832;\n","recurrence_sentence":"The row polynomials satisfy (1+(9*n-3)*x+(9*n^2+16*n-12)*x^2+(24*n^2-6*n-6)*x^3+(16*n^2-16*n+4)*x^4)*P(n,x) = (1+(9*n+9)*x+(9*n^2+61*n+33)*x^2+(51*n^2+162*n+63)*x^3+(106*n^2+210*n+66)*x^4+(96*n^2+132*n+36)*x^5+(32*n^2+32*n+8)*x^6)*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[((1 + (9*n + 9)*x + (9*n^2 + 61*n + 33)*x^2 + (51*n^2 + 162*n + 63)*x^3 + (106*n^2 + 210*n + 66)*x^4 + (96*n^2 + 132*n + 36)*x^5 + (32*n^2 + 32*n + 8)*x^6)*P[n - 1, x])/(1 + (9*n - 3)*x + (9*n^2 + 16*n - 12)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (16*n^2 - 16*n + 4)*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] = ((1 + (9*n + 9)*x + (9*n^2 + 61*n + 33)*x^2 + (51*n^2 + 162*n + 63)*x^3 + (106*n^2 + 210*n + 66)*x^4 + (96*n^2 + 132*n + 36)*x^5 + (32*n^2 + 32*n + 8)*x^6)*v[i - 1])/(1 + (9*n - 3)*x + (9*n^2 + 16*n - 12)*x^2 + (24*n^2 - 6*n - 6)*x^3 + (16*n^2 - 16*n + 4)*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-eed4f744ce063213","steps":["(\u03b8+1)\u00b2"],"edges":["e-4c485da9704e1764"],"text":"A200536 \u2192 (\u03b8+1)\u00b2"},"headline":"A200536 \u2192 (\u03b8+1)\u00b2","chain":null}