{"id":"n-fa77c904daa26e1a","revision":1,"first_index":1,"fingerprint":"f6-dd135a606f8f7ab8","name":"basis_map_A008277 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["6","6","1"],["16","29","11","1"],["44","127","83","17","1"],["120","528","532","184","24","1"],["328","2129","3115","1620","351","32","1"],["896","8419","17245","12667","4054","606","41","1"],["2448","32868","91988","91795","40717","8871","974","51","1"],["6688","127205","478219","631543","373088","111110","17597","1483","62","1"],["18272","489355","2440901","4186897","3206168","1253672","268549","32377","2164","74","1"],["49920","1874640","12293040","27011320","26293976","13132730","3653950","590886","56120","3051","87","1"],["136384","7160177","61299759","170725026","208199081","130204958","45766394","9531866","1206103","92659","4181","101","1"],["372608","27290755","303406013","1062274529","1604694688","1237856873","538700440","140525112","22744982","2315033","146926","5594","116","1"],["1017984","103861788","1493274020","6529703547","12110362169","11389326586","6043796503","1934979636","389567828","50435697","4220976","225142","7333","132","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","57","272","1389","7576","43929","269713","1746996","11898430","84959721","634326690","4940333670","40050779342"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162234,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-3ab95efc77e6349f"]},{"id":162235,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-3ab95efc77e6349f"]},{"id":162236,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-3ab95efc77e6349f"]}],"region_exact":{},"incoming_edges":[{"id":"e-3ab95efc77e6349f","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-fa77c904daa26e1a","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"31fcc57df4b353cc245bd940dfd6ac92fd7d17c13c939e30d4c49c7180f3a16d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    2,      1;\n   3:    6,      6,      1;\n   4:   16,     29,     11,      1;\n   5:   44,    127,     83,     17,      1;\n   6:  120,    528,    532,    184,     24,      1;\n   7:  328,   2129,   3115,   1620,    351,     32,     1;\n   8:  896,   8419,  17245,  12667,   4054,    606,    41,    1;\n   9: 2448,  32868,  91988,  91795,  40717,   8871,   974,   51,  1;\n  10: 6688, 127205, 478219, 631543, 373088, 111110, 17597, 1483, 62, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (2 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (2 + x)*v[i - 1] + x*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["basis A008277"],"edges":["e-3ab95efc77e6349f"],"text":"A063967 \u2192 basis A008277"},"headline":"A063967 \u2192 basis A008277","chain":null}