{"id":"n-4636bdf7e3e77f22","revision":0,"first_index":0,"fingerprint":"f6-6d5781c4134cee14","name":"basis_map_A013611 of Triangular matrix arising in enumeration of catafusenes, read by rows.","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":true,"rows":[["1"],["2","4"],["8","40","48"],["32","256","672","576"],["128","1408","5760","10368","6912"],["512","7168","39936","110592","152064","82944"],["2048","34816","245760","921600","1935360","2156544","995328"],["8192","163840","1400832","6635520","18800640","31850496","29859840","11943936"],["32768","753664","7569408","43352064","154828800","353009664","501645312","406093824","143327232"],["131072","3407872","39321600","264241152","1139539968","3269984256","6242697216","7644119040","5446434816","1719926784"],["524288","15204352","198180864","1528823808","7729053696","26754416640","64210599936","105488842752","113515167744","72236924928","20639121408"],["2097152","67108864","975175680","8493465600","49262100480","199766310912","577895399424","1192482570240","1719926784000","1651129712640","949399584768","247669456896"],["8388608","293601280","4706009088","45675970560","298969989120","1390210449408","4708777328640","11704675074048","21189497978880","27243640258560","23611154890752","12383472844800","2972033482752"],["33554432","1275068416","22347251712","239175991296","1743991603200","9148481667072","35517634707456","103324028239872","225214092804096","363248536780800","421368302665728","332867750068224","160489808068608","35664401793024"],["134217728","5502926848","104689827840","1224870985728","9846078308352","57521825906688","251852318834688","839507729448960","2140744710094848","4155563260772352","6044455652032512","6387890632261632","4636372233093120","2068535303995392","427972821516288"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","96","1536","24576","393216","6291456","100663296","1610612736","25769803776","412316860416","6597069766656","105553116266496","1688849860263936","27021597764222976"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-12"],["4","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204854,"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-091a70a69c7d9100"]},{"id":204855,"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-091a70a69c7d9100"]},{"id":204856,"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-091a70a69c7d9100"]},{"id":204857,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-091a70a69c7d9100"]}],"region_exact":{},"incoming_edges":[{"id":"e-091a70a69c7d9100","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-4636bdf7e3e77f22","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":"9eaa637e6f3f8ceb356af8d4a5f77ae59a4aa95fc1462573714298a38cf4030c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-685ab5842fbad7f4","role":"sequence","name":"Triangular matrix arising in enumeration of catafusenes, read by rows.","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-03ce471a34308b05","kind":"fingerprint_related","node_a_id":"n-6edd791871a0d803","node_b_id":"n-4636bdf7e3e77f22","fingerprint":"f6-6d5781c4134cee14","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6d5781c4134cee14"},"other_id":"n-6edd791871a0d803","other":{"id":"n-6edd791871a0d803","name":"Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.","first_index":0,"fingerprint":"f6-6d5781c4134cee14","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","5","6"]],"card_rows":[["1"],["1","2"],["1","5","6"],["1","8","21","18"]],"row_sums_preview":["1","3","12","48","192","768","3072","12288"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A193722"],"derivation":{"root":"n-6edd791871a0d803","steps":[],"edges":[],"text":"A193722"},"headline":"A193722: Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.","chain":null}},{"id":"l-152f7410752826cd","kind":"fingerprint_related","node_a_id":"n-8006e68a699fe151","node_b_id":"n-4636bdf7e3e77f22","fingerprint":"f6-6d5781c4134cee14","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6d5781c4134cee14"},"other_id":"n-8006e68a699fe151","other":{"id":"n-8006e68a699fe151","name":"basis_map_A007318 of Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.","first_index":1,"fingerprint":"f6-6d5781c4134cee14","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["6","5","1"]],"card_rows":[["1"],["2","1"],["6","5","1"],["18","21","8","1"]],"row_sums_preview":["1","3","12","48","192","768","3072","12288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f2a2a3d7ea29bcbf","steps":["basis A007318"],"edges":["e-92a0376fde876b5f"],"text":"A200139 \u2192 basis A007318"},"headline":"A200139 \u2192 basis A007318","chain":null}}],"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:      2,       4;\n  2:      8,      40,       48;\n  3:     32,     256,      672,       576;\n  4:    128,    1408,     5760,     10368,       6912;\n  5:    512,    7168,    39936,    110592,     152064,      82944;\n  6:   2048,   34816,   245760,    921600,    1935360,    2156544,     995328;\n  7:   8192,  163840,  1400832,   6635520,   18800640,   31850496,   29859840,   11943936;\n  8:  32768,  753664,  7569408,  43352064,  154828800,  353009664,  501645312,  406093824,  143327232;\n  9: 131072, 3407872, 39321600, 264241152, 1139539968, 3269984256, 6242697216, 7644119040, 5446434816, 1719926784;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (4*n-4+(12*n-12)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (12*n - 12)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (12*n - 12)*x)*v[i - 1])/(n - 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-685ab5842fbad7f4","steps":["basis A013611"],"edges":["e-091a70a69c7d9100"],"text":"A038763 \u2192 basis A013611"},"headline":"A038763 \u2192 basis A013611","chain":null}