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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d51e1b12c78931f6b38185e418007aa9d6c00e676f74a87a85e4225348b2cc79","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-c914b69838ac70ec","role":"sequence","name":"Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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":[{"id":"l-73695d9a68859a91","kind":"fingerprint_related","node_a_id":"n-c914b69838ac70ec","node_b_id":"n-800b2507fc1b82d7","fingerprint":"f6-9568e61f1cbcfb1d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9568e61f1cbcfb1d"},"other_id":"n-c914b69838ac70ec","other":{"id":"n-c914b69838ac70ec","name":"Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","first_index":0,"fingerprint":"f6-9568e61f1cbcfb1d","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","2"]],"card_rows":[["1"],["1"],["1","2"],["1","6"]],"row_sums_preview":["1","1","3","7","25","81","331","1303"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A059344"],"derivation":{"root":"n-c914b69838ac70ec","steps":[],"edges":[],"text":"A059344"},"headline":"A059344: Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","chain":null}}],"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:  0;\n  2:  2;\n  3:  6;\n  4: 12,   24;\n  5: 20,  120;\n  6: 30,  360,   360;\n  7: 42,  840,  2520;\n  8: 56, 1680, 10080,  6720;\n  9: 72, 3024, 30240, 60480;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n+(2*n^2-6*n)*x)*P(n-1,x) - 4*n*x^2*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (2*n^2 - 6*n)*x)*P[n - 1, x] - 4*n*x^2*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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