{"id":"n-5896c729e0a05938","revision":0,"first_index":0,"fingerprint":"f6-70d79adb4c85c3c6","name":"basis_map_A007318 of The (1,2)-Pascal triangle (or Lucas triangle) 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":[["2"],["3","2"],["6","7","2"],["12","20","11","2"],["24","52","42","15","2"],["48","128","136","72","19","2"],["96","304","400","280","110","23","2"],["192","704","1104","960","500","156","27","2"],["384","1600","2912","3024","1960","812","210","31","2"],["768","3584","7424","8960","6944","3584","1232","272","35","2"],["1536","7936","18432","25344","22848","14112","6048","1776","342","39","2"],["3072","17408","44800","69120","71040","51072","26208","9600","2460","420","43","2"],["6144","37888","107008","183040","211200","173184","103488","45408","14520","3300","506","47","2"],["12288","81920","251904","473088","605440","557568","380160","194304","74448","21120","4312","600","51","2"],["24576","176128","585728","1198080","1683968","1720576","1317888","768768","343200","116688","29744","5512","702","55","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","5","15","45","135","405","1215","3645","10935","32805","98415","295245","885735","2657205","7971615"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["3","2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195212,"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-dab4d9040d19e888"]},{"id":195213,"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-dab4d9040d19e888"]},{"id":195214,"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-dab4d9040d19e888"]},{"id":195215,"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-dab4d9040d19e888"]}],"region_exact":{},"incoming_edges":[{"id":"e-dab4d9040d19e888","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-5896c729e0a05938","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":"52dbb4e885bc1d2a6b03ddf76918f4f613223481a79f63e020dfa6667ad0f97e","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-b636fb6adf08770c","role":"sequence","name":"The (1,2)-Pascal triangle (or Lucas triangle) 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-a1fde5749ca467ca","kind":"fingerprint_related","node_a_id":"n-5896c729e0a05938","node_b_id":"n-29d29565124ec96c","fingerprint":"f6-70d79adb4c85c3c6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-70d79adb4c85c3c6"},"other_id":"n-29d29565124ec96c","other":{"id":"n-29d29565124ec96c","name":"basis_map_A013609 of Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.","first_index":0,"fingerprint":"f6-70d79adb4c85c3c6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["4","6"],["4","14","12"]],"card_rows":[["3"],["4","6"],["4","14","12"],["4","22","40","24"]],"row_sums_preview":["3","10","30","90","270","810","2430","7290"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-863aa63e001c0689","steps":["basis A013609"],"edges":["e-ef8a3dba9c03efe1"],"text":"A136157 \u2192 basis A013609"},"headline":"A136157 \u2192 basis A013609","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:   2;\n  1:   3,    2;\n  2:   6,    7,    2;\n  3:  12,   20,   11,    2;\n  4:  24,   52,   42,   15,    2;\n  5:  48,  128,  136,   72,   19,    2;\n  6:  96,  304,  400,  280,  110,   23,    2;\n  7: 192,  704, 1104,  960,  500,  156,   27,   2;\n  8: 384, 1600, 2912, 3024, 1960,  812,  210,  31,  2;\n  9: 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*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] = 2; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*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-b636fb6adf08770c","steps":["basis A007318"],"edges":["e-dab4d9040d19e888"],"text":"A029635 \u2192 basis A007318"},"headline":"A029635 \u2192 basis A007318","chain":null}
