{"id":"n-60783defa9dbe6a9","revision":1,"first_index":0,"fingerprint":"f6-1bc3b9efcd292160","name":"basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","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":[["1"],["15","4"],["225","120","16"],["3375","2700","720","64"],["50625","54000","21600","3840","256"],["759375","1012500","540000","144000","19200","1024"],["11390625","18225000","12150000","4320000","864000","92160","4096"],["170859375","318937500","255150000","113400000","30240000","4838400","430080","16384"],["2562890625","5467500000","5103000000","2721600000","907200000","193536000","25804800","1966080","65536"],["38443359375","92264062500","98415000000","61236000000","24494400000","6531840000","1161216000","132710400","8847360","262144"],["576650390625","1537734375000","1845281250000","1312200000000","612360000000","195955200000","43545600000","6635520000","663552000","39321600","1048576"],["8649755859375","25372617187500","33830156250000","27064125000000","14434200000000","5388768000000","1437004800000","273715200000","36495360000","3244032000","173015040","4194304"],["129746337890625","415188281250000","608942812500000","541282500000000","324769500000000","138568320000000","43110144000000","9853747200000","1642291200000","194641920000","15571353600","754974720","16777216"],["1946195068359375","6746809570312500","10794895312500000","10555008750000000","7036672500000000","3377602800000000","1200925440000000","320246784000000","64049356800000","9488793600000","1012137984000","73610035200","3271557120","67108864"],["29192926025390625","108986923828125000","188910667968750000","201504712500000000","147770122500000000","78810732000000000","31524292800000000","9607403520000000","2241727488000000","398529331200000","53137244160000","5152702464000","343513497600","14092861440","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","19","361","6859","130321","2476099","47045881","893871739","16983563041","322687697779","6131066257801","116490258898219","2213314919066161","42052983462257059","799006685782884121"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["15","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168203,"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-8d978ffb07d0025e"]},{"id":168204,"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-8d978ffb07d0025e"]},{"id":168205,"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-8d978ffb07d0025e"]}],"region_exact":{},"incoming_edges":[{"id":"e-8d978ffb07d0025e","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-60783defa9dbe6a9","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":"e61e2a2370d3559a3d56b6e9d79e79b3fa4033e8086299ec6731e8b036652973","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-a679863398df5985","role":"sequence","name":"Triangle of coefficients in expansion of (14 + x)^n.","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-19d906c663056de3","kind":"oeis_match","node_a_id":"n-60783defa9dbe6a9","node_b_id":null,"fingerprint":null,"oeis_id":"A001029","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001029","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001029","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-60783defa9dbe6a9","oeis_id":"A001029","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":26438}],"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:          15,           4;\n  2:         225,         120,          16;\n  3:        3375,        2700,         720,          64;\n  4:       50625,       54000,       21600,        3840,         256;\n  5:      759375,     1012500,      540000,      144000,       19200,       1024;\n  6:    11390625,    18225000,    12150000,     4320000,      864000,      92160,       4096;\n  7:   170859375,   318937500,   255150000,   113400000,    30240000,    4838400,     430080,     16384;\n  8:  2562890625,  5467500000,  5103000000,  2721600000,   907200000,  193536000,   25804800,   1966080,   65536;\n  9: 38443359375, 92264062500, 98415000000, 61236000000, 24494400000, 6531840000, 1161216000, 132710400, 8847360, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (15+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(15 + 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; for(n = 1, N, my(i = n + 1); v[i] = (15 + 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-a679863398df5985","steps":["basis A013611"],"edges":["e-8d978ffb07d0025e"],"text":"A147716 \u2192 basis A013611"},"headline":"A147716 \u2192 basis A013611","chain":null}