{"id":"n-249359630d4f905d","revision":0,"first_index":0,"fingerprint":"f6-e8b28356a6da96c8","name":"basis_map_A013610 of basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^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"],["23","45"],["529","2070","2025"],["12167","71415","139725","91125"],["279841","2190060","6427350","8383500","4100625"],["6436343","62964225","246381750","482051250","471571875","184528125"],["148035889","1737812610","8500170375","22174357500","32538459375","25464881250","8303765625"],["3404825447","46631305035","273705486075","892517889375","1746230653125","2049922940625","1336906265625","373669453125"],["78310985281","1225737160920","8393634906300","32844658329000","80326610043750","125728607025000","122995376437500","68755179375000","16815125390625"],["1801152661463","31715949038805","248211775086300","1133140712350500","3325521655811250","6506455413543750","8486680974187500","7116161065312500","3480730955859375","756680642578125"],["41426511213649","810518697658350","7136088533731125","37231766262945000","127478330139431250","299296949023012500","487984156015781250","545572348340625000","400284059923828125","174036547792968750","34050628916015625"],["952809757913927","20506123050756255","200603377670441625","1177454608065635625","4607431075039443750","12620354683803693750","24691998294398531250","34507451032544531250","33757289053576171875","22015623295810546875","8614809115751953125","1532278301220703125"],["21914624432020321","514517269273520580","5536653223704188850","36108607980679492500","158956372088860809375","497602556104259925000","1135831921542332437500","1904811296996458125000","2329252944696755859375","2025437343214570312500","1188843657973769531250","422908811136914062500","68952523554931640625"],["504036361936467383","12820055292731887785","150496301262504769650","1079647378622316825750","5280883917174375778125","18597895534396714696875","48516249220165342687500","94923096300323496562500","139289326092866000390625","151401441405289130859375","118488084578052363281250","63224867264968652343750","20616804542924560546875","3102863559971923828125"],["11592836324538749809","317542908019974451290","4038317417210544652275","31604223265126001626500","170044462133014900055625","665391373563971348043750","1952779031111655043171875","4366462429814880841875000","7475193833650475354296875","9750252826500620027343750","9538290808533215244140625","6786135753106635351562500","3319305531410854248046875","999122066310959472656250","139628860198736572265625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","68","4624","314432","21381376","1453933568","98867482624","6722988818432","457163239653376","31087100296429568","2113922820157210624","143746751770690322432","9774779120406941925376","664684980187672050925568","45198578652761699462938624"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["23","45"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203719,"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-afefdd7b4c52a451"]},{"id":203720,"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-afefdd7b4c52a451"]},{"id":203721,"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-afefdd7b4c52a451"]}],"region_exact":{},"incoming_edges":[{"id":"e-afefdd7b4c52a451","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-249359630d4f905d","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":"84acad6de97fdc5343c8a1bcb8ab9635313ac941b90d26de210bd3a8a3027069","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-a6cd3f3657ee08fd","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (3+5x)^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-86190e2a214bf2ea","kind":"fingerprint_related","node_a_id":"n-0bb7630b749d3bfa","node_b_id":"n-249359630d4f905d","fingerprint":"f6-e8b28356a6da96c8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e8b28356a6da96c8"},"other_id":"n-0bb7630b749d3bfa","other":{"id":"n-0bb7630b749d3bfa","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","first_index":0,"fingerprint":"f6-e8b28356a6da96c8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["529","2070","2025"],["12167","71415","139725","91125"],["279841","2190060","6427350","8383500","4100625"]],"card_rows":[["529","2070","2025"],["12167","71415","139725","91125"],["279841","2190060","6427350","8383500","4100625"],["6436343","62964225","246381750","482051250","471571875","184528125"]],"row_sums_preview":["4624","314432","21381376","1453933568","98867482624","6722988818432","457163239653376","31087100296429568"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-344e4bfa9933088a","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-9947ebaa7f991114","e-c85d4b4708d455d6"],"text":"A038337 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038337 \u2192 basis A013609 \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:            23,             45;\n  2:           529,           2070,            2025;\n  3:         12167,          71415,          139725,            91125;\n  4:        279841,        2190060,         6427350,          8383500,          4100625;\n  5:       6436343,       62964225,       246381750,        482051250,        471571875,        184528125;\n  6:     148035889,     1737812610,      8500170375,      22174357500,      32538459375,      25464881250,       8303765625;\n  7:    3404825447,    46631305035,    273705486075,     892517889375,    1746230653125,    2049922940625,    1336906265625,     373669453125;\n  8:   78310985281,  1225737160920,   8393634906300,   32844658329000,   80326610043750,  125728607025000,  122995376437500,   68755179375000,   16815125390625;\n  9: 1801152661463, 31715949038805, 248211775086300, 1133140712350500, 3325521655811250, 6506455413543750, 8486680974187500, 7116161065312500, 3480730955859375, 756680642578125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+45*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[(23 + 45*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] = (23 + 45*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-c44f7114310405ad","steps":["basis A013610","basis A013610"],"edges":["e-c78a98c6ca91439c","e-afefdd7b4c52a451"],"text":"A013622 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A013622 \u2192 basis A013610 \u2192 basis A013610","chain":null}