{"id":"n-b6b8b7173a216e50","revision":0,"first_index":0,"fingerprint":"f6-b7ed2a977b2f213e","name":"basis_map_A013609 of basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^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"],["13","16"],["169","416","256"],["2197","8112","9984","4096"],["28561","140608","259584","212992","65536"],["371293","2284880","5624320","6922240","4259840","1048576"],["4826809","35644128","109674240","179978240","166133760","81788928","16777216"],["62748517","540602608","1996071168","4094504960","5039390720","3721396224","1526726656","268435456"],["815730721","8031810176","34598566912","85165703168","131024158720","129008402432","79389786112","27917287424","4294967296"],["10604499373","117465223824","578290332672","1660731211776","3065965314048","3773495771136","3096201658368","1633161314304","502511173632","68719476736"],["137858491849","1696719899680","9397217905920","30842151075840","66429248471040","98110890049536","100626553896960","70770323619840","32663226286080","8933531975680","1099511627776"],["1792160394037","24263094565424","149311351171840","551303450480640","1357054647336960","2338309546180608","2877919441453056","2530039069409280","1556947119636480","638747536261120","157230162771968","17592186044416"],["23298085122481","344094795655104","2329257078280704","9555926474997760","26462565623070720","52110898457739264","74825905477779456","78937218965569536","60720937665822720","33214871885578240","12263952696213504","2744381022928896","281474976710656"],["302875106592253","4846001705476048","35785858748130816","161495157427462144","496908176699883520","1100842729919741952","1806511146534961152","2223398334196875264","2052367693104807936","1403328337165680640","690869335220027392","231900196437491712","47569271064100864","4503599627370496"],["3937376385699289","67844023876664672","542752191013317376","2672010786527100928","9043728815937880064","22261486316154781696","41098128583670366208","57808356689118756864","62255153357512507392","51081151472830775296","31434554752511246336","14068611917207830528","4328803666833178624","819655132181430272","72057594037927936"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","29","841","24389","707281","20511149","594823321","17249876309","500246412961","14507145975869","420707233300201","12200509765705829","353814783205469041","10260628712958602189","297558232675799463481"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["13","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200535,"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-a1ab9f6dbc9c9996","e-b1b26c6a917e70a3"]},{"id":200536,"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-a1ab9f6dbc9c9996","e-b1b26c6a917e70a3"]},{"id":200537,"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-a1ab9f6dbc9c9996","e-b1b26c6a917e70a3"]}],"region_exact":{},"incoming_edges":[{"id":"e-a1ab9f6dbc9c9996","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-b6b8b7173a216e50","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":"9a5f20799bcd9452ef29dfef3b4054852d6cff4d5d2f1332d2c19448e7fec504","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-b8c0cc94a00a4dd8","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^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]"}]},{"id":"e-b1b26c6a917e70a3","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-b6b8b7173a216e50","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":"fddfc4943d8d21864313fd59a147776ee3609f40692df4a5bdd033cb0fa05851","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-210f23d82ddfabd8","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^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-15ab2edef909c4dd","kind":"candidate_equal","node_a_id":"n-576df5f1b88caa47","node_b_id":"n-b6b8b7173a216e50","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-576df5f1b88caa47","other":{"id":"n-576df5f1b88caa47","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","first_index":0,"fingerprint":"f6-b7ed2a977b2f213e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["13","16"],["169","416","256"]],"card_rows":[["1"],["13","16"],["169","416","256"],["2197","8112","9984","4096"]],"row_sums_preview":["1","29","841","24389","707281","20511149","594823321","17249876309"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-81e6fb4f9f226d77","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-00e3cfe53baaea9b","e-307bf19bad8908a0"],"text":"A038327 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A038327 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-77e1d43751029249","kind":"fingerprint_related","node_a_id":"n-4070f46c6317d3ae","node_b_id":"n-b6b8b7173a216e50","fingerprint":"f6-b7ed2a977b2f213e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b7ed2a977b2f213e"},"other_id":"n-4070f46c6317d3ae","other":{"id":"n-4070f46c6317d3ae","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b7ed2a977b2f213e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["169","416","256"],["2197","8112","9984","4096"],["28561","140608","259584","212992","65536"]],"card_rows":[["169","416","256"],["2197","8112","9984","4096"],["28561","140608","259584","212992","65536"],["371293","2284880","5624320","6922240","4259840","1048576"]],"row_sums_preview":["841","24389","707281","20511149","594823321","17249876309","500246412961","14507145975869"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4397e742863fdbe2","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-6056f3721ff524be","e-1daa5e0124e0bbc8"],"text":"A038305 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A038305 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:          13,           16;\n  2:         169,          416,          256;\n  3:        2197,         8112,         9984,          4096;\n  4:       28561,       140608,       259584,        212992,         65536;\n  5:      371293,      2284880,      5624320,       6922240,       4259840,       1048576;\n  6:     4826809,     35644128,    109674240,     179978240,     166133760,      81788928,      16777216;\n  7:    62748517,    540602608,   1996071168,    4094504960,    5039390720,    3721396224,    1526726656,     268435456;\n  8:   815730721,   8031810176,  34598566912,   85165703168,  131024158720,  129008402432,   79389786112,   27917287424,   4294967296;\n  9: 10604499373, 117465223824, 578290332672, 1660731211776, 3065965314048, 3773495771136, 3096201658368, 1633161314304, 502511173632, 68719476736;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (13+16*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[(13 + 16*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] = (13 + 16*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-366b2525683554ec","steps":["basis A013611","basis A013611"],"edges":["e-b30dba0e9f8b339c","e-a1ab9f6dbc9c9996"],"text":"A038279 \u2192 basis A013611 \u2192 basis A013611"},"headline":"A038279 \u2192 basis A013611 \u2192 basis A013611","chain":null}