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Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"9fd96a495a5f4b1444ce3bfbdc662a85e42655597dd0f6726f6fe0fbd6e5d30b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1faa73e065080941","kind":"oeis_match","node_a_id":"n-55697ddae4d22d60","node_b_id":null,"fingerprint":null,"oeis_id":"A123160","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A123160","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3730f1cc8fbbc131","kind":"oeis_match","node_a_id":"n-55697ddae4d22d60","node_b_id":null,"fingerprint":null,"oeis_id":"A123164","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A123164","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c020b81c7d0f0be8","kind":"oeis_match","node_a_id":"n-55697ddae4d22d60","node_b_id":null,"fingerprint":null,"oeis_id":"A002003","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A002003","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A123160","relation":"identification"},{"oeis_id":"A002003","relation":"match_row_sums"},{"oeis_id":"A123164","relation":"match_row_sums"},{"oeis_id":"A123160","relation":"match_triangle"}],"bfiles":[{"node_id":"n-55697ddae4d22d60","oeis_id":"A123160","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":20526},{"node_id":"n-55697ddae4d22d60","oeis_id":"A123164","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":25,"oeis_terms_available":25,"term_count":200,"raw_size":15943}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  1;\n   3: 1,  4,    3;\n   4: 1,  9,   18,    10;\n   5: 1, 16,   60,    80,    35;\n   6: 1, 25,  150,   350,   350,    126;\n   7: 1, 36,  315,  1120,  1890,   1512,    462;\n   8: 1, 49,  588,  2940,  7350,   9702,   6468,   1716;\n   9: 1, 64, 1008,  6720, 23100,  44352,  48048,  27456,   6435;\n  10: 1, 81, 1620, 13860, 62370, 162162, 252252, 231660, 115830, 24310;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = -(n+(4*n-6)*x)*P(n-1,x) - (n-5+(4*n-14)*x)*P(n-2,x) + (n-4)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 4*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[3, x_] := 1 + 4*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + (4*n - 6)*x)*P[n - 1, x] - (n - 5 + (4*n - 14)*x)*P[n - 2, x] + (n - 4)*P[n - 3, x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; v[3] = 1 + 4*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(n + (4*n - 6)*x)*v[i - 1] - (n - 5 + (4*n - 14)*x)*v[i - 2] + (n - 4)*v[i - 3])/(-n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A123160","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  1;\n  2: 1,  4,    3;\n  3: 1,  9,   18,    10;\n  4: 1, 16,   60,    80,    35;\n  5: 1, 25,  150,   350,   350,    126;\n  6: 1, 36,  315,  1120,  1890,   1512,    462;\n  7: 1, 49,  588,  2940,  7350,   9702,   6468,   1716;\n  8: 1, 64, 1008,  6720, 23100,  44352,  48048,  27456,   6435;\n  9: 1, 81, 1620, 13860, 62370, 162162, 252252, 231660, 115830, 24310;\n","recurrence_sentence":"The row polynomials satisfy -n*P(n,x) = -(n+1+(4*n-2)*x)*P(n-1,x) - (n-4+(4*n-10)*x)*P(n-2,x) + (n-3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 1 + 4*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[2, x_] := 1 + 4*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (4*n - 2)*x)*P[n - 1, x] - (n - 4 + (4*n - 10)*x)*P[n - 2, x] + (n - 3)*P[n - 3, x])/(-n)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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