{"id":"n-3a5b707c177aa910","revision":0,"first_index":0,"fingerprint":"f6-0205823d5b324a40","name":"shift_x_plus_one of 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":[["324","144","16"],["5832","3888","864","64"],["104976","93312","31104","4608","256"],["1889568","2099520","933120","207360","23040","1024"],["34012224","45349632","25194240","7464960","1244160","110592","4096"],["612220032","952342272","634894848","235146240","52254720","6967296","516096","16384"],["11019960576","19591041024","15237476352","6772211712","1881169920","334430208","37158912","2359296","65536"],["198359290368","396718580736","352638738432","182849716224","60949905408","13544423424","2006581248","191102976","10616832","262144"],["3570467226624","7934371614720","7934371614720","4701849845760","1828497162240","487599243264","90296156160","11466178560","955514880","47185920","1048576"],["64268410079232","157100557971456","174556175523840","116370783682560","51720348303360","16090775027712","3575727783936","567575838720","63063982080","4671406080","207618048","4194304"],["1156831381426176","3084883683803136","3770413391314944","2792898808381440","1396449404190720","496515343712256","128726200221696","24519276232704","3405455032320","336341237760","22422749184","905969664","16777216"],["20822964865671168","60155231834161152","80206975778881536","65353832116125696","36307684508958720","14523073803583488","4303132978839552","956251773075456","159375295512576","19675962408960","1748974436352","105998450688","3925868544","67108864"],["374813367582081024","1166086032477585408","1684346491356512256","1497196881205788672","914953649625759744","406646066500337664","135548688833445888","34425063830716416","6693762411528192","991668505411584","110185389490176","8903869857792","494659436544","16911433728","268435456"],["6746640616477458432","22488802054924861440","34982580974327562240","33686929827130245120","22457953218086830080","10979443795509116928","4066460665003376640","1161845904286679040","258187978730373120","44625082743521280","5950011032469504","601011215400960","44519349288960","2283043553280","72477573120","1073741824"],["121439531096594251776","431784999454557339648","719641665757595566080","746295060785654661120","538990877234083921920","287461801191511425024","117114067152097247232","37179068937173729280","9294767234293432320","1836003404304875520","285600529558536192","34618246007095296","3205393148805120","219172181114880","10436770529280","309237645312","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792","26559922791424","584318301411328","12855002631049216","282810057883082752","6221821273427820544","136880068015412051968","3011361496339065143296"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["324","144","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["18","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192766,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-0ca963c00c6b2bae"]},{"id":192767,"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-0ca963c00c6b2bae","e-2ce118e2f9073f6b"]},{"id":192768,"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-0ca963c00c6b2bae","e-2ce118e2f9073f6b"]},{"id":192769,"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-0ca963c00c6b2bae"]},{"id":192770,"position":4,"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-2ce118e2f9073f6b"]}],"region_exact":{},"incoming_edges":[{"id":"e-2ce118e2f9073f6b","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3a5b707c177aa910","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":"8943395464aad544c8531a4b602bee97a0a8e1f092e00fff2854a7e26ff5cfb8","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-3208539f9ba783e8","role":"sequence","name":"basis_map_A013609 of 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]"}]},{"id":"e-0ca963c00c6b2bae","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3a5b707c177aa910","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"f9e24235dc4d625313aa0be597c837e3bd6b5346972ae9367205ba0cd951da9a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3208539f9ba783e8","role":"sequence","name":"basis_map_A013609 of 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-5917a9ca9eacbe78","kind":"fingerprint_related","node_a_id":"n-da07574fa3209102","node_b_id":"n-3a5b707c177aa910","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-da07574fa3209102","other":{"id":"n-da07574fa3209102","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["36","8"]],"card_rows":[["2"],["36","8"],["486","216","24"],["5832","3888","864","64"]],"row_sums_preview":["0","2","44","726","10648","146410","1932612","24801854"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013609","D"],"edges":["e-0da8e8e5bf86fd8d","e-0e1fda911995b99e"],"text":"A038279 \u2192 basis A013609 \u2192 D"},"headline":"A038279 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-6efcab153b4b5cc3","kind":"fingerprint_related","node_a_id":"n-20a6b63b1c7531be","node_b_id":"n-3a5b707c177aa910","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-20a6b63b1c7531be","other":{"id":"n-20a6b63b1c7531be","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","2"],["81","36","4"]],"card_rows":[["1"],["9","2"],["81","36","4"],["729","486","108","8"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013609"],"edges":["e-0da8e8e5bf86fd8d"],"text":"A038279 \u2192 basis A013609"},"headline":"A038279 \u2192 basis A013609","chain":null}},{"id":"l-d67a04f03f7aa4a8","kind":"fingerprint_related","node_a_id":"n-3a5b707c177aa910","node_b_id":"n-4675eb44920357da","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-4675eb44920357da","other":{"id":"n-4675eb44920357da","name":"basis_map_A013609 of basis_map_A013609 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","4"],["324","144","16"]],"card_rows":[["1"],["18","4"],["324","144","16"],["5832","3888","864","64"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["basis A013609","basis A013609"],"edges":["e-f16669f3777d4fa6","e-cc970da74fc6993c"],"text":"A027467 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A027467 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-f842e3010af9fbe6","kind":"fingerprint_related","node_a_id":"n-24d208fb12b74cc6","node_b_id":"n-3a5b707c177aa910","fingerprint":"f6-0205823d5b324a40","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0205823d5b324a40"},"other_id":"n-24d208fb12b74cc6","other":{"id":"n-24d208fb12b74cc6","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","first_index":0,"fingerprint":"f6-0205823d5b324a40","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","9"],["4","36","81"]],"card_rows":[["1"],["2","9"],["4","36","81"],["8","108","486","729"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-366b2525683554ec","steps":["basis A013609","reverse"],"edges":["e-0da8e8e5bf86fd8d","e-fa090f76810df478"],"text":"A038279 \u2192 basis A013609 \u2192 reverse"},"headline":"A038279 \u2192 basis A013609 \u2192 reverse","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:            324,             144,              16;\n  1:           5832,            3888,             864,              64;\n  2:         104976,           93312,           31104,            4608,            256;\n  3:        1889568,         2099520,          933120,          207360,          23040,           1024;\n  4:       34012224,        45349632,        25194240,         7464960,        1244160,         110592,          4096;\n  5:      612220032,       952342272,       634894848,       235146240,       52254720,        6967296,        516096,        16384;\n  6:    11019960576,     19591041024,     15237476352,      6772211712,     1881169920,      334430208,      37158912,      2359296,       65536;\n  7:   198359290368,    396718580736,    352638738432,    182849716224,    60949905408,    13544423424,    2006581248,    191102976,    10616832,     262144;\n  8:  3570467226624,   7934371614720,   7934371614720,   4701849845760,  1828497162240,   487599243264,   90296156160,  11466178560,   955514880,   47185920,   1048576;\n  9: 64268410079232, 157100557971456, 174556175523840, 116370783682560, 51720348303360, 16090775027712, 3575727783936, 567575838720, 63063982080, 4671406080, 207618048, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 144*x + 16*x^2.","mathematica":"P[0, x_] := 324 + 144*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(18 + 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] = 324 + 144*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (18 + 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","x \u21a6 x+1"],"edges":["e-ccf7b0982517b62a","e-0ca963c00c6b2bae"],"text":"A038328 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038328 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}