{"id":"n-3d02bb5acf6eb220","revision":1,"first_index":0,"fingerprint":"f6-d19c49978a21a529","name":"one_plus_derivative of Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["7","6","1"],["13","21","9","1"],["21","52","42","12","1"],["31","105","130","70","15","1"],["43","186","315","260","105","18","1"],["57","301","651","735","455","147","21","1"],["73","456","1204","1736","1470","728","196","24","1"],["91","657","2052","3612","3906","2646","1092","252","27","1"],["111","910","3285","6840","9030","7812","4410","1560","315","30","1"],["133","1221","5005","12045","18810","19866","14322","6930","2145","385","33","1"],["157","1596","7326","20020","36135","45144","39732","24552","10395","2860","462","36","1"],["183","2041","10374","31746","65065","93951","97812","73788","39897","15015","3718","546","39","1"],["211","2562","14287","48412","111111","182182","219219","195624","129129","62062","21021","4732","637","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","14","44","128","352","928","2368","5888","14336","34304","80896","188416","434176","991232"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","0","1"],["-1","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["1","3","3","1"],["1","3","2","0"],["1","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163391,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-31ecd8005cc9acb4"]},{"id":163392,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-31ecd8005cc9acb4"]},{"id":163393,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-31ecd8005cc9acb4"]}],"region_exact":{},"incoming_edges":[{"id":"e-31ecd8005cc9acb4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-3d02bb5acf6eb220","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"fd64c2aa858d4b075fd3da228e6fec4a335ac65da3884a42f84e2559d1681b2c","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-f24dc5f77f658331","role":"sequence","name":"Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","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-c04b6f466e930e42","kind":"oeis_match","node_a_id":"n-3d02bb5acf6eb220","node_b_id":null,"fingerprint":null,"oeis_id":"A007466","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007466","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007466","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-3d02bb5acf6eb220","oeis_id":"A007466","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":27,"oeis_terms_available":27,"term_count":200,"raw_size":7613}],"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:  3,   1;\n  2:  7,   6,    1;\n  3: 13,  21,    9,    1;\n  4: 21,  52,   42,   12,    1;\n  5: 31, 105,  130,   70,   15,    1;\n  6: 43, 186,  315,  260,  105,   18,    1;\n  7: 57, 301,  651,  735,  455,  147,   21,   1;\n  8: 73, 456, 1204, 1736, 1470,  728,  196,  24,  1;\n  9: 91, 657, 2052, 3612, 3906, 2646, 1092, 252, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n+1+2*n*x+x^2)*P(n,x) = (n^2+n+1+(n^2+3*n+3)*x+(2*n+3)*x^2+x^3)*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[((n^2 + n + 1 + (n^2 + 3*n + 3)*x + (2*n + 3)*x^2 + x^3)*P[n - 1, x])/(n^2 - n + 1 + 2*n*x + x^2)]];\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] = ((n^2 + n + 1 + (n^2 + 3*n + 3)*x + (2*n + 3)*x^2 + x^3)*v[i - 1])/(n^2 - n + 1 + 2*n*x + x^2)); 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-c8c19eba9d68881f","steps":["D+1","D+1"],"edges":["e-5c65a1a70531c253","e-31ecd8005cc9acb4"],"text":"A007318 \u2192 D+1 \u2192 D+1"},"headline":"A007318 \u2192 D+1 \u2192 D+1","chain":null}
