{"id":"n-fb13dff3dfb4d666","revision":0,"first_index":0,"fingerprint":"f6-78576ef9a6880520","name":"shift_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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"],["5","2"],["31","25","5"],["215","261","105","14"],["1597","2590","1568","420","42"],["12425","25220","20400","8220","1650","132"],["99955","243675","246735","132825","40095","6435","429"],["824675","2346995","2854775","1923845","775775","187187","25025","1430"],["6939769","22582780","32072656","25965940","13107094","4224220","848848","97240","4862"],["59334605","217299942","352928160","333648588","202333950","81624816","21905520","3771144","377910","16796"],["513972967","2092108525","3824616705","4135199250","2928361800","1419217800","476723520","109593900","16502070","1469650","58786"],["4501041935","20158633649","40964510125","49857215100","40381341000","22853726280","9222166800","2653451460","533482950","71379770","5720330","208012"],["39784038517","194419150650","434744618880","588206443000","536308314300","347155102800","163586657520","56541731400","14226887070","2541460900","305955936","22287000","742900"],["354455513105","1876867987592","4579838904160","6819052842888","6912012950700","5036822875656","2714660554320","1095664788504","331165430310","74032837736","11898286400","1301857960","86919300","2674440"],["3179928556219","18136051812655","47955206521131","77921664681315","86923433415819","70420190735895","42727180038543","19724166498465","6961412224767","1869332339875","375948668095","54911267775","5506337655","339319575","9694845"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","61","595","6217","68047","770149","8939707","105843409","1273241431","15517824973","191202877411","2377843390873","29807864423071","376255282112629"],"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":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["10","4"],["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["12","10","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-fdd65bc6f0fa1af3","recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1"],["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","2"],["10","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182000,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-3, -2)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, -2)","region_exact":null,"edges":["e-601c9dcf57fce44e"]},{"id":182001,"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-601c9dcf57fce44e","e-62bbdecd427fd7aa"]},{"id":182002,"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-601c9dcf57fce44e","e-62bbdecd427fd7aa"]},{"id":182003,"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-601c9dcf57fce44e","e-62bbdecd427fd7aa"]},{"id":182004,"position":4,"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-62bbdecd427fd7aa"]}],"region_exact":{},"incoming_edges":[{"id":"e-62bbdecd427fd7aa","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-fb13dff3dfb4d666","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":"b1f9bb1638bcf3dc1e1bc6514374cfa3da24637d09cff5966a9714a2d1c65bda","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-07af2e6e3c9f0c77","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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-fdd65bc6f0fa1af3","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-fb13dff3dfb4d666","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":"d04418f30bf9027e75aa4d3c27c47dc75e39416ef61bf3940e214034ae9a6138","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-601c9dcf57fce44e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-fb13dff3dfb4d666","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":"7c9ed3dd5786a6a1c301a86e89cefb3321e221735a668bd37e33b986f62a9f90","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-2"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-07af2e6e3c9f0c77","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, -1)"}]}],"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:        1;\n  1:        5,         2;\n  2:       31,        25,         5;\n  3:      215,       261,       105,        14;\n  4:     1597,      2590,      1568,       420,        42;\n  5:    12425,     25220,     20400,      8220,      1650,      132;\n  6:    99955,    243675,    246735,    132825,     40095,     6435,      429;\n  7:   824675,   2346995,   2854775,   1923845,    775775,   187187,    25025,    1430;\n  8:  6939769,  22582780,  32072656,  25965940,  13107094,  4224220,   848848,   97240,   4862;\n  9: 59334605, 217299942, 352928160, 333648588, 202333950, 81624816, 21905520, 3771144, 377910, 16796;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (5*n+10+(2*n+4)*x)*P(n-1,x) + (12+10*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[Together[((5*n + 10 + (2*n + 4)*x)*P[n - 1, x] + (12 + 10*x + 2*x^2)*D[P[n - 1, x], x])/(n + 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] = ((5*n + 10 + (2*n + 4)*x)*v[i - 1] + (12 + 10*x + 2*x^2)*deriv(v[i - 1], x))/(n + 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-66e2c3448245b0b7","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-190c283dd8b47413","e-601c9dcf57fce44e"],"text":"A033282 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A033282 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}