{"id":"n-3dddab772a6867d2","revision":1,"first_index":0,"fingerprint":"f6-5104a571b27debd0","name":"basis_map_A013609 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","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"],["4","6"],["9","22","12"],["22","76","80","24"],["53","236","368","232","48"],["128","698","1440","1384","608","96"],["309","1994","5116","6616","4496","1504","192"],["746","5560","17056","27728","25600","13312","3584","384"],["1801","15224","54336","106416","124384","88320","36992","8320","768"],["4348","41102","167296","383344","542656","489920","280832","98176","18944","1536"],["10497","109726","501580","1316368","2189216","2402240","1755136","840320","251648","42496","3072"],["25342","290244","1472112","4353832","8320512","10758144","9575424","5850752","2399232","627712","94208","6144"],["61181","761892","4245744","13971416","30170640","44937984","47226752","35202944","18432256","6599680","1531904","206848","12288"],["147704","1987074","12067872","43732376","105312288","177616416","215421184","189860992","121371136","55490048","17612800","3672064","450560","24576"],["356589","5153810","33879420","134063400","356202800","671136672","923177920","940220800","711302400","397186560","160936960","45840384","8671232","974848","49152"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","43","202","937","4354","20227","93970","436561","2028154","9422299","43773658","203361529","944767090","4389152947"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166380,"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-30bb9f6b8a15c298"]},{"id":166381,"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-30bb9f6b8a15c298"]},{"id":166382,"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-30bb9f6b8a15c298"]},{"id":166383,"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-30bb9f6b8a15c298"]}],"region_exact":{},"incoming_edges":[{"id":"e-30bb9f6b8a15c298","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3dddab772a6867d2","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":"4d8ad0f75823ccc7633f845d002ce9111b07b6441392bc9779def1759014b349","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-14dfd91463cae288","role":"sequence","name":"Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","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:    1;\n  1:    4,     6;\n  2:    9,    22,     12;\n  3:   22,    76,     80,     24;\n  4:   53,   236,    368,    232,     48;\n  5:  128,   698,   1440,   1384,    608,     96;\n  6:  309,  1994,   5116,   6616,   4496,   1504,    192;\n  7:  746,  5560,  17056,  27728,  25600,  13312,   3584,   384;\n  8: 1801, 15224,  54336, 106416, 124384,  88320,  36992,  8320,   768;\n  9: 4348, 41102, 167296, 383344, 542656, 489920, 280832, 98176, 18944, 1536;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 2, x] + (2 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 2] + (2 + 2*x)*v[i - 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-14dfd91463cae288","steps":["basis A013609"],"edges":["e-30bb9f6b8a15c298"],"text":"A102662 \u2192 basis A013609"},"headline":"A102662 \u2192 basis A013609","chain":null}
