{"id":"n-33acd11b57e0c597","revision":0,"first_index":0,"fingerprint":"f6-977fd250db0d872d","name":"basis_map_A013610 of Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.","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":[["3"],["4","9"],["4","21","27"],["4","33","90","81"],["4","45","189","351","243"],["4","57","324","918","1296","729"],["4","69","495","1890","4050","4617","2187"],["4","81","702","3375","9720","16767","16038","6561"],["4","93","945","5481","19845","45927","66339","54675","19683"],["4","105","1224","8316","36288","105462","204120","253692","183708","59049"],["4","117","1539","11988","61236","214326","520506","866052","944784","610173","177147"],["4","129","1890","16605","97200","398034","1163484","2427570","3542940","3444525","2007666","531441"],["4","141","2277","22275","147015","689634","2357586","5918022","10825650","14073345","12341241","6554439","1594323"],["4","153","2700","29106","213840","1130679","4426488","12990780","28579716","46550295","54561276","43578162","21257640","4782969"],["4","165","3159","37206","301158","1772199","7818525","26270244","67552056","132289443","194212161","207261990","151992126","68555889","14348907"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["3","13","52","208","832","3328","13312","53248","212992","851968","3407872","13631488","54525952","218103808","872415232"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"],["4","9"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-3"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201264,"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-dd73a88d9dc9f038"]},{"id":201265,"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-dd73a88d9dc9f038"]},{"id":201266,"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-dd73a88d9dc9f038"]},{"id":201267,"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-dd73a88d9dc9f038"]}],"region_exact":{},"incoming_edges":[{"id":"e-dd73a88d9dc9f038","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-33acd11b57e0c597","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":"214da3bbff6668a06544947587e81b84ed15c80656f65ce6094c74f2bd86814f","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-863aa63e001c0689","role":"sequence","name":"Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.","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: 3;\n  1: 4,   9;\n  2: 4,  21,   27;\n  3: 4,  33,   90,   81;\n  4: 4,  45,  189,  351,   243;\n  5: 4,  57,  324,  918,  1296,    729;\n  6: 4,  69,  495, 1890,  4050,   4617,   2187;\n  7: 4,  81,  702, 3375,  9720,  16767,  16038,   6561;\n  8: 4,  93,  945, 5481, 19845,  45927,  66339,  54675,  19683;\n  9: 4, 105, 1224, 8316, 36288, 105462, 204120, 253692, 183708, 59049;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n-1+(3*n-3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 4 + 9*x.","mathematica":"P[0, x_] := 3;\nP[1, x_] := 4 + 9*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (3*n - 3)*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] = 3; v[2] = 4 + 9*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (3*n - 3)*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-863aa63e001c0689","steps":["basis A013610"],"edges":["e-dd73a88d9dc9f038"],"text":"A136157 \u2192 basis A013610"},"headline":"A136157 \u2192 basis A013610","chain":null}