{"id":"n-6a6677e3386580cd","revision":1,"first_index":0,"fingerprint":"f6-df7cb345936f8559","name":"basis_map_A013609 of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["2","2"],["4","6"],["10","24","12"],["26","80","60"],["76","300","360","120"],["232","1092","1680","840"],["764","4256","8400","6720","1680"],["2620","16704","39312","40320","15120"],["9496","68760","191520","252000","151200","30240"],["35696","288200","918720","1441440","1108800","332640"],["140152","1253472","4538160","8426880","8316000","3991680","665280"],["568504","5568576","22479600","47773440","56216160","34594560","8648640"],["2390480","25507664","114065952","275315040","383423040","302702400","121080960","17297280"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","10","46","166","856","3844","21820","114076","703216","4125496","27331624","175849480","1241782816"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-2"],["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166355,"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-ffb008f602bb5117"]},{"id":166356,"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-ffb008f602bb5117"]},{"id":166357,"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-ffb008f602bb5117"]},{"id":166358,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-ffb008f602bb5117"]}],"region_exact":{},"incoming_edges":[{"id":"e-ffb008f602bb5117","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-6a6677e3386580cd","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"58bf59ec39dfdb327cec0b84441fa0700615a3f85167c04594356f7dbf2ccf11","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-97b51734d39ede6c","role":"sequence","name":"Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","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-4fa149ad1b113bce","kind":"oeis_match","node_a_id":"n-6a6677e3386580cd","node_b_id":null,"fingerprint":null,"oeis_id":"A115327","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A115327","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A115327","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-6a6677e3386580cd","oeis_id":"A115327","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":22580}],"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;\n  2:    2,     2;\n  3:    4,     6;\n  4:   10,    24,    12;\n  5:   26,    80,    60;\n  6:   76,   300,   360,   120;\n  7:  232,  1092,  1680,   840;\n  8:  764,  4256,  8400,  6720,  1680;\n  9: 2620, 16704, 39312, 40320, 15120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+(2*n-2)*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + (2*n - 2)*x)*P[n - 2, x] + P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (2*n - 2)*x)*v[i - 2] + v[i - 1]); 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-97b51734d39ede6c","steps":["basis A013609"],"edges":["e-ffb008f602bb5117"],"text":"A100861 \u2192 basis A013609"},"headline":"A100861 \u2192 basis A013609","chain":null}
