{"id":"n-16217e05bf18203f","revision":1,"first_index":0,"fingerprint":"f6-dbff0e5fad25cf92","name":"basis_map_A013609 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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"],["13","36","24"],["43","174","228","96"],["142","756","1476","1248","384"],["469","3096","8028","10200","6336","1536"],["1549","12204","39456","66912","62688","30720","6144"],["5116","46842","181404","384816","482448","357120","144384","24576"],["16897","176292","795444","2025888","3183120","3157248","1929216","663552","98304"],["55807","653538","3366588","10007064","18904992","23526240","19275264","10020864","2998272","393216"],["184318","2393928","13861944","47104320","103978128","155722176","160174464","111685632","50503680","13369344","1572864"],["608761","8684208","55835208","213500976","539270784","944415264","1169739840","1024326144","621293568","248512512","58982400","6291456"],["2010601","31250232","220892256","938671968","2670003024","5354007360","7758499392","8183972352","6235060224","3345088512","1199308800","257949696","25165824"],["6640564","111694830","860881320","4024756320","12730969680","28764991008","47750097984","58949720448","54111015936","36466606080","17534287872","5695340544","1119879168","100663296"],["21932293","396918180","3312816000","16898250720","58839281280","147912865152","276776772096","391550524416","420734934528","341659090944","206358773760","89879740416","26682064896","4831838208","402653184"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","73","541","4006","29665","219673","1626706","12045961","89201845","660550798","4891461121","36221880241","268227545050","1986258456073"],"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":[["3","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166503,"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-346e78903a645c68"]},{"id":166504,"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-346e78903a645c68"]},{"id":166505,"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-346e78903a645c68"]},{"id":166506,"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-346e78903a645c68"]}],"region_exact":{},"incoming_edges":[{"id":"e-346e78903a645c68","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-16217e05bf18203f","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":"aa01fa51c091ea344590147171026c049106355cc0092081f1e6605edc5eb7f6","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-e7331b0d50084c15","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:    13,     36,      24;\n  3:    43,    174,     228,       96;\n  4:   142,    756,    1476,     1248,      384;\n  5:   469,   3096,    8028,    10200,     6336,     1536;\n  6:  1549,  12204,   39456,    66912,    62688,    30720,     6144;\n  7:  5116,  46842,  181404,   384816,   482448,   357120,   144384,    24576;\n  8: 16897, 176292,  795444,  2025888,  3183120,  3157248,  1929216,   663552,   98304;\n  9: 55807, 653538, 3366588, 10007064, 18904992, 23526240, 19275264, 10020864, 2998272, 393216;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-2,x) + (3+4*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] + (3 + 4*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] + (3 + 4*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-e7331b0d50084c15","steps":["basis A013609"],"edges":["e-346e78903a645c68"],"text":"A249251 \u2192 basis A013609"},"headline":"A249251 \u2192 basis A013609","chain":null}