{"id":"n-3f2d3a36a2f92f4a","revision":0,"first_index":0,"fingerprint":"f6-28c423639b74b313","name":"basis_map_A013611 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.","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"],["2","4"],["7","28","16"],["26","156","240","64"],["124","992","2544","2240","256"],["680","6800","24608","38848","23296","1024"],["3952","47424","221568","507904","575232","261120","4096"],["23456","328384","1876992","5634560","9402880","8297472","3043328","16384"],["140224","2243584","15145216","56101888","123458560","161701888","116912128","36093952","65536"],["840320","15125760","117646848","517665792","1411909632","2447585280","2636390400","1614790656","431161344","262144"],["5039872","100797440","887021568","4515790848","14676566016","31611912192","45161840640","41293971456","21940862976","5165023232","1048576"],["30235136","665172992","6530797568","37733597184","142227259392","365729710080","650192879616","789532901376","626999820288","294165151744","61940432896","4194304"],["181402624","4353662976","47164698624","304756736000","1306101841920","3900897755136","8289429356544","12538681491456","13235355648000","9288066007040","3901070770176","743109033984","16777216"],["1088399360","28298383360","335227035648","2394479312896","11493504450560","39077933875200","96545549844480","174701642514432","229870913716224","214546665963520","134858418421760","51268886200320","8916536655872","67108864"],["6530363392","182850174976","2350930886656","18389504884736","97694253645824","372387557408768","1048202207821824","2206742058369024","3475620001480704","4045697372389376","3383676914106368","1925889910636544","668713640525824","106995084427264","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","51","486","6156","95256","1621296","28623456","511862976","9193378176","165359874816","2975752152576","53559185181696","964039211882496","17352549085556736"],"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"],["7","28","16"],["26","156","240","64"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["36","144","144"],["-12","-48","-48"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-24","-48"],["8","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205194,"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-6feb4d060c383157"]},{"id":205195,"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-6feb4d060c383157"]},{"id":205196,"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-6feb4d060c383157"]},{"id":205197,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-6feb4d060c383157"]}],"region_exact":{},"incoming_edges":[{"id":"e-6feb4d060c383157","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-3f2d3a36a2f92f4a","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":"0518ccbead76230a45ad84b5da04d18094a290cd10194a8fd38899c75e87d462","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fbab854f2622405b","role":"sequence","name":"Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.","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:      2,        4;\n  2:      7,       28,        16;\n  3:     26,      156,       240,        64;\n  4:    124,      992,      2544,      2240,        256;\n  5:    680,     6800,     24608,     38848,      23296,       1024;\n  6:   3952,    47424,    221568,    507904,     575232,     261120,       4096;\n  7:  23456,   328384,   1876992,   5634560,    9402880,    8297472,    3043328,      16384;\n  8: 140224,  2243584,  15145216,  56101888,  123458560,  161701888,  116912128,   36093952,     65536;\n  9: 840320, 15125760, 117646848, 517665792, 1411909632, 2447585280, 2636390400, 1614790656, 431161344, 262144;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(48*n-144)*x+(48*n-144)*x^2)*P(n-2,x) + (8*n-24+(16*n-48)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 7 + 28*x + 16*x^2, P(3,x) = 26 + 156*x + 240*x^2 + 64*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 7 + 28*x + 16*x^2;\nP[3, x_] := 26 + 156*x + 240*x^2 + 64*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (48*n - 144)*x + (48*n - 144)*x^2)*P[n - 2, x] + (8*n - 24 + (16*n - 48)*x)*P[n - 1, x])/(n - 3)]];\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; v[3] = 7 + 28*x + 16*x^2; v[4] = 26 + 156*x + 240*x^2 + 64*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (48*n - 144)*x + (48*n - 144)*x^2)*v[i - 2] + (8*n - 24 + (16*n - 48)*x)*v[i - 1])/(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-fbab854f2622405b","steps":["basis A013611"],"edges":["e-6feb4d060c383157"],"text":"A146987 \u2192 basis A013611"},"headline":"A146987 \u2192 basis A013611","chain":null}