{"id":"n-8159f24e9c9fb552","revision":1,"first_index":0,"fingerprint":"f6-a5e5e7677b9b26e0","name":"shift_x_plus_one 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","1"],["4","3"],["10","12","3"],["26","40","15"],["76","150","90","15"],["232","546","420","105"],["764","2128","2100","840","105"],["2620","8352","9828","5040","945"],["9496","34380","47880","31500","9450","945"],["35696","144100","229680","180180","69300","10395"],["140152","626736","1134540","1053360","519750","124740","10395"],["568504","2784288","5619900","5971680","3513510","1081080","135135"],["2390480","12753832","28516488","34414380","23963940","9459450","1891890","135135"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","3","7","25","81","331","1303","5937","26785","133651","669351","3609673","19674097","113525595"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-47ee1dd4d233fe8f","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161461,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-de232437ce21276e"]},{"id":161462,"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-d700e7dd8b41ed69","e-de232437ce21276e"]},{"id":161463,"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-d700e7dd8b41ed69","e-de232437ce21276e"]},{"id":161464,"position":3,"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-d700e7dd8b41ed69"]},{"id":161465,"position":4,"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-d700e7dd8b41ed69"]}],"region_exact":{},"incoming_edges":[{"id":"e-d700e7dd8b41ed69","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-8159f24e9c9fb552","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":"1d9602f937ab7f407a61a87cc1292e35c4d2f3eb3d3a84f83aec6cd7b7394098","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]"}]},{"id":"e-47ee1dd4d233fe8f","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-8159f24e9c9fb552","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"5e5aa98967c8d2d381f440c081e70c65a21c737993f10a37a6fb4fe24b1ccda7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-de232437ce21276e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-8159f24e9c9fb552","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"ee044dc82ec34ea1757ba18d207f45ef8fb6ce9fa43ea3540986dab9a6a07606","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"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-56f260873205bfac","kind":"fingerprint_related","node_a_id":"n-8159f24e9c9fb552","node_b_id":"n-e92431a37829c185","fingerprint":"f6-a5e5e7677b9b26e0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a5e5e7677b9b26e0"},"other_id":"n-e92431a37829c185","other":{"id":"n-e92431a37829c185","name":"mobius_x_over_x_plus_one of Exponential Riordan array (1, x(1+x/2)).","first_index":0,"fingerprint":"f6-a5e5e7677b9b26e0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","2"]],"card_rows":[["1"],["0","1"],["0","1","2"],["0","0","3","4"]],"row_sums_preview":["1","1","3","7","25","81","331","1303"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97b51734d39ede6c","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-01d50252f4aaa594","e-fc862e6fd76dbdc8"],"text":"A100861 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A100861 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1bcae3e82d9efc67","kind":"oeis_match","node_a_id":"n-8159f24e9c9fb552","node_b_id":null,"fingerprint":null,"oeis_id":"A047974","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":"A047974","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A047974","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-8159f24e9c9fb552","oeis_id":"A047974","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":27,"oeis_terms_available":27,"term_count":200,"raw_size":20934}],"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,    1;\n  3:    4,    3;\n  4:   10,   12,    3;\n  5:   26,   40,   15;\n  6:   76,  150,   90,   15;\n  7:  232,  546,  420,  105;\n  8:  764, 2128, 2100,  840, 105;\n  9: 2620, 8352, 9828, 5040, 945;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+(n-1)*x)*P(n-1,x) - (2+4*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + (n - 1)*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*D[P[n - 1, x], x]];\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 + (n - 1)*x)*v[i - 1] - (2 + 4*x + 2*x^2)*deriv(v[i - 1], x)); 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 A007318"],"edges":["e-d700e7dd8b41ed69"],"text":"A100861 \u2192 basis A007318"},"headline":"A100861 \u2192 basis A007318","chain":null}
