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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":"f7c783a65f34559e68eaa479e5657a702b6036aeb456d8bea780fd3639d699de","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-b30e7a5b4198051a","role":"sequence","name":"Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","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,    9;\n  2: 1,  24,   99,    96,     25;\n  3: 1,  36,  270,   720,    750,     324,      49;\n  4: 1,  48,  522,  2304,   4875,    5184,    2842,      768,       81;\n  5: 1,  60,  855,  5280,  17125,   31428,   33565,    21120,     7695,     1500,      121;\n  6: 1,  72, 1269, 10080,  44250,  117288,  195461,   208512,   143370,    63000,    17061,     2592,      169;\n  7: 1,  84, 1764, 17136,  95025,  331128,  761117,  1182912,  1258173,   919800,   459921,   154224,    33124,    4116,     225;\n  8: 1,  96, 2340, 26880, 180250,  780192, 2299472,  4753920,  7007715,  7428000,  5678288,  3120768,  1218490,  329280,   58500,   6144,   289;\n  9: 1, 108, 2997, 39744, 312750, 1619352, 5838546, 15151104, 28858923, 40810500, 43110243, 34089984, 20137026, 8816472, 2814750, 635904, 96237, 8748, 361;\n","recurrence_sentence":"The row polynomials satisfy (1+(9*n-3)*x+(9*n^2+8*n-6)*x^2+(12*n^2-3*n-3)*x^3+(4*n^2-4*n+1)*x^4)*P(n,x) = (1+(9*n+9)*x+(9*n^2+53*n+30)*x^2+(39*n^2+108*n+45)*x^3+(49*n^2+93*n+30)*x^4+(24*n^2+33*n+9)*x^5+(4*n^2+4*n+1)*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 + 53*n + 30)*x^2 + (39*n^2 + 108*n + 45)*x^3 + (49*n^2 + 93*n + 30)*x^4 + (24*n^2 + 33*n + 9)*x^5 + (4*n^2 + 4*n + 1)*x^6)*P[n - 1, x])/(1 + (9*n - 3)*x + (9*n^2 + 8*n - 6)*x^2 + (12*n^2 - 3*n - 3)*x^3 + (4*n^2 - 4*n + 1)*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 + 53*n + 30)*x^2 + (39*n^2 + 108*n + 45)*x^3 + (49*n^2 + 93*n + 30)*x^4 + (24*n^2 + 33*n + 9)*x^5 + (4*n^2 + 4*n + 1)*x^6)*v[i - 1])/(1 + (9*n - 3)*x + (9*n^2 + 8*n - 6)*x^2 + (12*n^2 - 3*n - 3)*x^3 + (4*n^2 - 4*n + 1)*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-b30e7a5b4198051a","steps":["(\u03b8+1)\u00b2"],"edges":["e-47750c013c3a531e"],"text":"A272866 \u2192 (\u03b8+1)\u00b2"},"headline":"A272866 \u2192 (\u03b8+1)\u00b2","chain":null}
