{"id":"n-6decca65eb6ed468","revision":1,"first_index":0,"fingerprint":"f6-3ac782335c346aee","name":"basis_map_A013609 of Triangle of Dyck paths counted by number of long interior inclines.","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":[["2"],["5","4"],["14","24","8"],["42","112","84","16"],["132","480","576","256","32"],["429","1980","3300","2400","720","64"],["1430","8008","17160","17600","8800","1920","128"],["4862","32032","84084","112112","80080","29568","4928","256"],["16796","127296","396032","652288","611520","326144","93184","12288","512"],["58786","503880","1813968","3564288","4158336","2935296","1223040","279552","29952","1024"],["208012","1989680","8139600","18604800","26046720","23030784","12794880","4300800","806400","71680","2048"],["742900","7845024","35956360","93799200","153489600","163722240","114605568","51701760","14361600","2252800","168960","4096"],["2674440","30904640","156900480","460241408","862952640","1080566784","916844544","523911168","196466688","45957120","6127616","393216","8192"],["9694845","121687020","677970540","2209681760","4674326800","6731030592","6731030592","4682456064","2234808576","709463040","141892608","16293888","905216","16384"],["35357670","479039400","2906172360","10422135360","24566461920","40034234240","46193347200","38010525696","22172806656","8997660672","2453907456","424919040","42491904","2064384","32768"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","9","46","254","1476","8893","55046","347922","2236060","14568122","95995404","638650108","4283948936","28941257925","196741156726"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["5","4"]],"recurrence":{"denominator":{"coeffs":[["3"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","12"],["0","0","-4"],["0","0","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","0"],["16","16"],["8","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166491,"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-ccbc6f9cbfe03222"]},{"id":166492,"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-ccbc6f9cbfe03222"]},{"id":166493,"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-ccbc6f9cbfe03222"]},{"id":166494,"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-ccbc6f9cbfe03222"]}],"region_exact":{},"incoming_edges":[{"id":"e-ccbc6f9cbfe03222","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-6decca65eb6ed468","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":"68ac05c58fd54b2c2a17f036747f635482230ec2243492706e64b96b7570494a","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-c963abfc0266d66c","role":"sequence","name":"Triangle of Dyck paths counted by number of long interior inclines.","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":[],"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:     2;\n  1:     5,      4;\n  2:    14,     24,       8;\n  3:    42,    112,      84,      16;\n  4:   132,    480,     576,     256,      32;\n  5:   429,   1980,    3300,    2400,     720,      64;\n  6:  1430,   8008,   17160,   17600,    8800,    1920,     128;\n  7:  4862,  32032,   84084,  112112,   80080,   29568,    4928,    256;\n  8: 16796, 127296,  396032,  652288,  611520,  326144,   93184,  12288,   512;\n  9: 58786, 503880, 1813968, 3564288, 4158336, 2935296, 1223040, 279552, 29952, 1024;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+7*n+3)*P(n,x) = -(8*n^2+4*n-12)*x^2*P(n-2,x) + (8*n^2+16*n+6+(8*n^2+16*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 4*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 + 4*n - 12)*x^2*P[n - 2, x] + (8*n^2 + 16*n + 6 + (8*n^2 + 16*n)*x)*P[n - 1, x])/(2*n^2 + 7*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^2 + 4*n - 12)*x^2*v[i - 2] + (8*n^2 + 16*n + 6 + (8*n^2 + 16*n)*x)*v[i - 1])/(2*n^2 + 7*n + 3)); 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-c963abfc0266d66c","steps":["basis A013609"],"edges":["e-ccbc6f9cbfe03222"],"text":"A108838 \u2192 basis A013609"},"headline":"A108838 \u2192 basis A013609","chain":null}