{"id":"n-28c47f7742a98cf2","revision":0,"first_index":0,"fingerprint":"f6-65e248b1aaac025a","name":"basis_map_A008277 of basis_map_A007318 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":false,"rows":[["1"],["9","4"],["81","88","16"],["729","1468","624","64"],["6561","22000","16480","3840","256"],["59049","312244","368160","146560","21760","1024"],["531441","4295368","7507696","4550400","1130240","116736","4096"],["4782969","57965548","144812304","125589184","46457600","7906304","602112","16384"],["43046721","772684000","2693671360","3216622080","1663796736","415113216","51494912","3014656","65536"],["387420489","10217078884","48883149120","78323749120","54461406720","18693470208","3359784960","317521920","14745600","262144"],["3486784401","134371707448","871881850576","1840331328000","1674830164480","759956262912","185646784512","25187450880","1874657280","70778880","1048576"],["31381059609","1760779334428","15359478289584","42134485290304","49232079424000","28738052282368","9166168940544","1674522820608","177610752000","10683678720","334495744","4194304"],["282429536481","23015655586000","268154248260640","946262104254720","1399339926761216","1030331833884672","417435784167424","98622020124672","13980132114432","1191208550400","59125006336","1560281088","16777216"],["2541865828329","300333240763924","4650684842774880","20944121182391680","38768546586049280","35476982889700352","17896708213063680","5318757881282560","967673497190400","109524913225728","7661959512064","319194923008","7197425664","67108864"],["22876792454961","3914499593244328","80262975290228656","458429284201324800","1052990149380798720","1183906690145507328","732499302589902848","268380874459709440","60949644909936640","8799315083919360","813535668994048","47565168967680","1687032954880","32883343360","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","185","2885","49137","908797","18135977","388132405","8859509217","214658589165","5497638817945","148285580562213","4198676045305297","124443863531240413","3851018433469722505"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["9","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197353,"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-c2461953ba27deea"]},{"id":197354,"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-c2461953ba27deea"]},{"id":197355,"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-c2461953ba27deea"]}],"region_exact":{},"incoming_edges":[{"id":"e-c2461953ba27deea","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-28c47f7742a98cf2","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7422bd1131c95f1cf7b6010f0d726324fe56333f3b552fc4deb857b84c02df2a","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-d04994959b21388c","role":"sequence","name":"basis_map_A007318 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-621991d758dbe231","kind":"fingerprint_related","node_a_id":"n-28c47f7742a98cf2","node_b_id":"n-47b7c605b3a36a4e","fingerprint":"f6-65e248b1aaac025a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-65e248b1aaac025a"},"other_id":"n-47b7c605b3a36a4e","other":{"id":"n-47b7c605b3a36a4e","name":"basis_map_A008277 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-65e248b1aaac025a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["729","792","144"],["19683","39636","16848","1728"],["531441","1782000","1334880","311040","20736"]],"card_rows":[["729","792","144"],["19683","39636","16848","1728"],["531441","1782000","1334880","311040","20736"],["14348907","75875292","89462880","35614080","5287680","248832"]],"row_sums_preview":["1665","77895","3980097","220837671","13221127233","848845569735","58127239972737","4225125010534695"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013611","basis A008277"],"edges":["e-a5d7492b5780a0c3","e-224ee59cff76398f"],"text":"A038329 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A038329 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-e548b3f4f8aa7820","kind":"fingerprint_related","node_a_id":"n-28c47f7742a98cf2","node_b_id":"n-1bf987212f59a9c9","fingerprint":"f6-65e248b1aaac025a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-65e248b1aaac025a"},"other_id":"n-1bf987212f59a9c9","other":{"id":"n-1bf987212f59a9c9","name":"basis_map_A008277 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-65e248b1aaac025a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1296","1408","256"],["46656","93952","39936","4096"],["1679616","5632000","4218880","983040","65536"]],"card_rows":[["1296","1408","256"],["46656","93952","39936","4096"],["1679616","5632000","4218880","983040","65536"],["60466176","319737856","376995840","150077440","22282240","1048576"]],"row_sums_preview":["2960","184640","12579072","930608128","74284961792","6359161323520","580616796045312","56271461198069760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013609","basis A008277"],"edges":["e-4eca5e8e987e069c","e-5fd8ac66802137f6"],"text":"A038334 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A038334 \u2192 basis A013609 \u2192 basis A008277","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:         9,           4;\n  2:        81,          88,          16;\n  3:       729,        1468,         624,          64;\n  4:      6561,       22000,       16480,        3840,         256;\n  5:     59049,      312244,      368160,      146560,       21760,        1024;\n  6:    531441,     4295368,     7507696,     4550400,     1130240,      116736,       4096;\n  7:   4782969,    57965548,   144812304,   125589184,    46457600,     7906304,     602112,     16384;\n  8:  43046721,   772684000,  2693671360,  3216622080,  1663796736,   415113216,   51494912,   3014656,    65536;\n  9: 387420489, 10217078884, 48883149120, 78323749120, 54461406720, 18693470208, 3359784960, 317521920, 14745600, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(9 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = (9 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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-2e0409c0014ec240","steps":["basis A007318","basis A008277"],"edges":["e-13c78408b476aecd","e-c2461953ba27deea"],"text":"A013611 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A013611 \u2192 basis A007318 \u2192 basis A008277","chain":null}
