{"id":"n-8ddb9ea84d7cbde5","revision":0,"first_index":0,"fingerprint":"f6-35d2a8c113de1376","name":"basis_map_A013610 of Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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"],["3","3"],["9","21","9"],["27","102","108","27"],["81","426","747","486","81"],["243","1641","4059","4455","2025","243"],["729","6015","19278","30159","23490","8019","729"],["2187","21324","84069","171639","192213","113967","30618","2187"],["6561","73812","345690","872856","1283283","1102248","520506","113724","6561"],["19683","250959","1361826","4102110","7492257","8492121","5848038","2270106","413343","19683"],["59049","841449","5192631","18185310","39764925","56210031","51313581","29222694","9546255","1476225","59049"],["177147","2790066","19297962","77075631","196519365","333604008","381110994","288900513","139217859","38972340","5196312","177147"],["531441","9167358","70249869","315227646","918835002","1823350014","2508352677","2386887426","1536809274","637847298","155239821","18068994","531441"],["1594323","29893557","251404821","1252390653","4109785452","9343243080","15059067453","17298529209","14020804512","7802262468","2828742345","605665593","62178597","1594323"],["4782969","96855123","886941828","4857553557","17727683085","45455071158","84091005117","113402535057","111206804553","78107064750","38090915577","12203125389","2321334288","212044959","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","39","264","1821","12666","88419","618204","4325241","30270126","211871199","1483039344","10381098261","72667156386","508668500379"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","-9","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201923,"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-367501c64407070e"]},{"id":201924,"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-367501c64407070e"]},{"id":201925,"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-367501c64407070e"]},{"id":201926,"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-367501c64407070e"]}],"region_exact":{},"incoming_edges":[{"id":"e-367501c64407070e","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-8ddb9ea84d7cbde5","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":"b7e22c712b525ff007add26fa37044c8dda96a09a877098f1c29b32e7a548024","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-9fa98aec39d32ffa","role":"sequence","name":"Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).","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:     3,      3;\n  2:     9,     21,       9;\n  3:    27,    102,     108,      27;\n  4:    81,    426,     747,     486,      81;\n  5:   243,   1641,    4059,    4455,    2025,     243;\n  6:   729,   6015,   19278,   30159,   23490,    8019,     729;\n  7:  2187,  21324,   84069,  171639,  192213,  113967,   30618,    2187;\n  8:  6561,  73812,  345690,  872856, 1283283, 1102248,  520506,  113724,   6561;\n  9: 19683, 250959, 1361826, 4102110, 7492257, 8492121, 5848038, 2270106, 413343, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3+9*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(3 + 9*x + 9*x^2)*P[n - 2, x] + (4 + 6*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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 9*x + 9*x^2)*v[i - 2] + (4 + 6*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-9fa98aec39d32ffa","steps":["basis A013610"],"edges":["e-367501c64407070e"],"text":"A104029 \u2192 basis A013610"},"headline":"A104029 \u2192 basis A013610","chain":null}