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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":"c66ff761233f190ca2949c33ecf358f2da4321e372db50ff3cab60c9bced5ec2","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-b60ac0f95ba69845","role":"sequence","name":"derivative of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=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: 0;\n 1: 3, 4;\n 2: 6, 44, 54, 16;\n 3: 9, 120, 405, 480, 225, 36;\n 4: 12, 232, 1296, 3120, 3600, 2088, 588, 64;\n 5: 15, 380, 2970, 10960, 21825, 24660, 16170, 6080, 1215, 100;\n 6: 18, 564, 5670, 28320, 81450, 143604, 159642, 113280, 51030, 14100, 2178, 144;\n 7: 21, 784, 9639, 60816, 229950, 559188, 905667, 994112, 745038, 380100, 129591, 28224, 3549, 196;\n 8: 24, 1040, 15120, 115360, 541800, 1689408, 3639720, 5536960, 6016680, 4692800, 2622312, 1038240, 283920, 50960, 5400, 256;\n 9: 27, 1332, 22356, 200160, 1124550, 4289544, 11600064, 22802112, 33056505, 35628300, 28645056, 17158176, 7601958, 2451960, 558900, 85248, 7803, 324;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3+(9*n^2-14*n+5)*x+(12*n^2-21*n+9)*x^2+(4*n^2-8*n+4)*x^3)*P(n,x) = (3*n+(9*n^2+13*n)*x+(39*n^2+18*n)*x^2+(49*n^2+13*n)*x^3+(24*n^2+3*n)*x^4+4*n^2*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + (9*n^2 + 13*n)*x + (39*n^2 + 18*n)*x^2 + (49*n^2 + 13*n)*x^3 + (24*n^2 + 3*n)*x^4 + 4*n^2*x^5)*P[n - 1, x])/(3*n - 3 + (9*n^2 - 14*n + 5)*x + (12*n^2 - 21*n + 9)*x^2 + (4*n^2 - 8*n + 4)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + (9*n^2 + 13*n)*x + (39*n^2 + 18*n)*x^2 + (49*n^2 + 13*n)*x^3 + (24*n^2 + 3*n)*x^4 + 4*n^2*x^5)*v[i - 1])/(3*n - 3 + (9*n^2 - 14*n + 5)*x + (12*n^2 - 21*n + 9)*x^2 + (4*n^2 - 8*n + 4)*x^3)); 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":["D","\u03b8+1"],"edges":["e-5fb206b09638d5b7","e-e27617ae3f087da7"],"text":"A272866 \u2192 D \u2192 \u03b8+1"},"headline":"A272866 \u2192 D \u2192 \u03b8+1","chain":null}