{"id":"n-3208539f9ba783e8","revision":1,"first_index":0,"fingerprint":"f6-8d10fbbacee008e8","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["196","112","16"],["2744","2352","672","64"],["38416","43904","18816","3584","256"],["537824","768320","439040","125440","17920","1024"],["7529536","12907776","9219840","3512320","752640","86016","4096"],["105413504","210827008","180708864","86051840","24586240","4214784","401408","16384"],["1475789056","3373232128","3373232128","1927561216","688414720","157351936","22478848","1835008","65536"],["20661046784","53128406016","60718178304","40478785536","17348050944","4956585984","944111616","115605504","8257536","262144"],["289254654976","826441871360","1062568120320","809575710720","404787855360","138784407552","33043906560","5394923520","578027520","36700160","1048576"],["4049565169664","12727204818944","18181721169920","15584332431360","8905332817920","3562133127168","1017752322048","207704555520","29672079360","2825912320","161480704","4194304"],["56693912375296","194379128143872","305452915654656","290907538718720","187011989176320","85491195052032","28497065017344","6978873065472","1246227333120","158251089920","13564379136","704643072","16777216"],["793714773254144","2948083443515392","5053857331740672","5294517204680704","3781798003343360","1944924687433728","740923690450944","211692482985984","45362674925568","7200424591360","822905667584","64122519552","3053453312","67108864"],["11112006825558016","44448027302232064","82546336418430976","94338670192492544","74123240865529856","42356137637445632","18152630416048128","5927389523607552","1481847380901888","282256643981312","40322377711616","4189337944064","299238424576","13153337344","268435456"],["155568095557812224","666720409533480960","1333440819066961920","1650926728368619520","1415080052887388160","889478890386358272","423561376374456320","155593974994698240","44455421427056640","9878982539345920","1693539863887872","219940242063360","20946689720320","1381100421120","56371445760","1073741824"],["2177953337809371136","9956358115699982336","21335053105071390720","28446737473428520960","26414827653897912320","18113024676958568448","9487774830787821568","3872561155423600640","1244751799957585920","316127441259069440","63225488251813888","9853322844438528","1173014624337920","103122164776960","6313601925120","240518168576","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368","3570467226624","64268410079232","1156831381426176","20822964865671168","374813367582081024","6746640616477458432","121439531096594251776"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["196","112","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["14","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166866,"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-ccf7b0982517b62a"]},{"id":166867,"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-ccf7b0982517b62a"]},{"id":166868,"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-ccf7b0982517b62a"]}],"region_exact":{},"incoming_edges":[{"id":"e-ccf7b0982517b62a","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3208539f9ba783e8","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":"7686b98796ba6d0870ae7a45a0330e209a62f2f93c283046fe4abddce19af688","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ea9b82d727693fb6","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","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-0507c04f7adeceb0","kind":"fingerprint_related","node_a_id":"n-6a4f4798749d2440","node_b_id":"n-3208539f9ba783e8","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-6a4f4798749d2440","other":{"id":"n-6a4f4798749d2440","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","2"],["49","28","4"]],"card_rows":[["1"],["7","2"],["49","28","4"],["343","294","84","8"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013609"],"edges":["e-4c464713c83c5f6b"],"text":"A038255 \u2192 basis A013609"},"headline":"A038255 \u2192 basis A013609","chain":null}},{"id":"l-30d5228d2c7b7968","kind":"oeis_match","node_a_id":"n-3208539f9ba783e8","node_b_id":null,"fingerprint":null,"oeis_id":"A001027","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001027","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001027","relation":"match_row_sums"}],"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:           196,            112,             16;\n  1:          2744,           2352,            672,             64;\n  2:         38416,          43904,          18816,           3584,           256;\n  3:        537824,         768320,         439040,         125440,         17920,          1024;\n  4:       7529536,       12907776,        9219840,        3512320,        752640,         86016,          4096;\n  5:     105413504,      210827008,      180708864,       86051840,      24586240,       4214784,        401408,        16384;\n  6:    1475789056,     3373232128,     3373232128,     1927561216,     688414720,     157351936,      22478848,      1835008,       65536;\n  7:   20661046784,    53128406016,    60718178304,    40478785536,   17348050944,    4956585984,     944111616,    115605504,     8257536,     262144;\n  8:  289254654976,   826441871360,  1062568120320,   809575710720,  404787855360,  138784407552,   33043906560,   5394923520,   578027520,   36700160,   1048576;\n  9: 4049565169664, 12727204818944, 18181721169920, 15584332431360, 8905332817920, 3562133127168, 1017752322048, 207704555520, 29672079360, 2825912320, 161480704, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 112*x + 16*x^2.","mathematica":"P[0, x_] := 196 + 112*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(14 + 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] = 196 + 112*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (14 + 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-ea9b82d727693fb6","steps":["basis A013609"],"edges":["e-ccf7b0982517b62a"],"text":"A038328 \u2192 basis A013609"},"headline":"A038328 \u2192 basis A013609","chain":null}
