{"id":"n-5b6560967f09bc16","revision":0,"first_index":0,"fingerprint":"f6-249d425e0def4c4f","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","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"],["18","20"],["324","720","400"],["5832","19440","21600","8000"],["104976","466560","777600","576000","160000"],["1889568","10497600","23328000","25920000","14400000","3200000"],["34012224","226748160","629856000","933120000","777600000","345600000","64000000"],["612220032","4761711360","15872371200","29393280000","32659200000","21772800000","8064000000","1280000000"],["11019960576","97955205120","380936908800","846526464000","1175731200000","1045094400000","580608000000","184320000000","25600000000"],["198359290368","1983592903680","8815968460800","22856214528000","38093690880000","42326323200000","31352832000000","14929920000000","4147200000000","512000000000"],["3570467226624","39671858073600","198359290368000","587731230720000","1142810726400000","1523747635200000","1410877440000000","895795200000000","373248000000000","92160000000000","10240000000000"],["64268410079232","785502789857280","4363904388096000","14546347960320000","32325217689600000","50283671961600000","55870746624000000","44341862400000000","24634368000000000","9123840000000000","2027520000000000","204800000000000"],["1156831381426176","15424418419015680","94260334782873600","349112351047680000","872780877619200000","1551610449100800000","2011346878464000000","1915568455680000000","1330255872000000000","656916480000000000","218972160000000000","44236800000000000","4096000000000000"],["20822964865671168","300776159170805760","2005174394472038400","8169229014515712000","22692302818099200000","45384605636198400000","67236452794368000000","74707169771520000000","62255974809600000000","38429614080000000000","17079828480000000000","5175705600000000000","958464000000000000","81920000000000000"],["374813367582081024","5830430162387927040","42108662283912806400","187149610150723584000","571846031016099840000","1270768957813555200000","2117948263022592000000","2689458111774720000000","2614750942003200000000","1936852549632000000000","1076029194240000000000","434759270400000000000","120766464000000000000","20643840000000000000","1638400000000000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","38","1444","54872","2085136","79235168","3010936384","114415582592","4347792138496","165216101262848","6278211847988224","238572050223552512","9065737908494995456","344498040522809827328","13090925539866773438464"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["18","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193057,"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-510d3f765aca8c0c","e-58472537e585bdda"]},{"id":193058,"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-510d3f765aca8c0c","e-58472537e585bdda"]},{"id":193059,"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-510d3f765aca8c0c","e-58472537e585bdda"]}],"region_exact":{},"incoming_edges":[{"id":"e-510d3f765aca8c0c","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-5b6560967f09bc16","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":"0348ec40549d1bff67f4fc7e8a40a2296547a018958d66f3cd0e13bcd393b5b7","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-09320ea6fe78bcb6","role":"sequence","name":"basis_map_A013609 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]"}]},{"id":"e-58472537e585bdda","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-5b6560967f09bc16","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"a3acfa556b1fce1c5587ca7b36be2b75870353b10eaaf1c2a37e679e4a47f5c3","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-76276d62adb9bc90","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^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-0703feeeefbf22be","kind":"fingerprint_related","node_a_id":"n-45d245d96cab6cbb","node_b_id":"n-5b6560967f09bc16","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-45d245d96cab6cbb","other":{"id":"n-45d245d96cab6cbb","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","10"],["81","180","100"]],"card_rows":[["1"],["9","10"],["81","180","100"],["729","2430","2700","1000"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["x \u21a6 x/(x+1)"],"edges":["e-b395db61f9c972bd"],"text":"A038291 \u2192 x \u21a6 x/(x+1)"},"headline":"A038291 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1b230ea3da81d7c6","kind":"fingerprint_related","node_a_id":"n-5b6560967f09bc16","node_b_id":"n-a0d5dc3c44ae9aed","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-a0d5dc3c44ae9aed","other":{"id":"n-a0d5dc3c44ae9aed","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["27","30"],["729","1620","900"]],"card_rows":[["1"],["27","30"],["729","1620","900"],["19683","65610","72900","27000"]],"row_sums_preview":["1","57","3249","185193","10556001","601692057","34296447249","1954897493193"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013610","basis A013609"],"edges":["e-cc1debec4a411a9c","e-a50e49aa95060606"],"text":"A038271 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038271 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-3455a31165cfe69a","kind":"fingerprint_related","node_a_id":"n-76276d62adb9bc90","node_b_id":"n-5b6560967f09bc16","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-76276d62adb9bc90","other":{"id":"n-76276d62adb9bc90","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","18"],["400","720","324"]],"card_rows":[["1"],["20","18"],["400","720","324"],["8000","21600","19440","5832"]],"row_sums_preview":["1","38","1444","54872","2085136","79235168","3010936384","114415582592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4e096b5929fd3f2e","steps":["basis A013609"],"edges":["e-66940ce826511afb"],"text":"A038323 \u2192 basis A013609"},"headline":"A038323 \u2192 basis A013609","chain":null}},{"id":"l-3ce5ed24ce939d66","kind":"fingerprint_related","node_a_id":"n-5b6560967f09bc16","node_b_id":"n-fbce10918920af94","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-fbce10918920af94","other":{"id":"n-fbce10918920af94","name":"basis_map_A013610 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["729","1620","900"],["19683","65610","72900","27000"],["531441","2361960","3936600","2916000","810000"]],"card_rows":[["729","1620","900"],["19683","65610","72900","27000"],["531441","2361960","3936600","2916000","810000"],["14348907","79716150","177147000","196830000","109350000","24300000"]],"row_sums_preview":["3249","185193","10556001","601692057","34296447249","1954897493193","111429157112001","6351461955384057"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a5302200d285579d","steps":["basis A013609","basis A013610"],"edges":["e-2f341f062a4ee4be","e-73e1a3f1e761636b"],"text":"A038331 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038331 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-5594c2c894c04c77","kind":"fingerprint_related","node_a_id":"n-5b6560967f09bc16","node_b_id":"n-fd7d4197f6940d35","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-fd7d4197f6940d35","other":{"id":"n-fd7d4197f6940d35","name":"basis_map_A013610 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","27"],["900","1620","729"]],"card_rows":[["1"],["30","27"],["900","1620","729"],["27000","72900","65610","19683"]],"row_sums_preview":["1","57","3249","185193","10556001","601692057","34296447249","1954897493193"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1","basis A013610"],"edges":["e-62ade03cd9f25f04","e-a361b3d68f92db8c"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A013610"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 basis A013610","chain":null}},{"id":"l-73ded9a669991f90","kind":"fingerprint_related","node_a_id":"n-5b6560967f09bc16","node_b_id":"n-010e63e80d598424","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-010e63e80d598424","other":{"id":"n-010e63e80d598424","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1296","2880","1600"],["46656","155520","172800","64000"],["1679616","7464960","12441600","9216000","2560000"]],"card_rows":[["1296","2880","1600"],["46656","155520","172800","64000"],["1679616","7464960","12441600","9216000","2560000"],["60466176","335923200","746496000","829440000","460800000","102400000"]],"row_sums_preview":["5776","438976","33362176","2535525376","192699928576","14645194571776","1113034787454976","84590643846578176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013611","basis A013609"],"edges":["e-fd91849598d32af1","e-29559957c78e1247"],"text":"A038319 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038319 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-8ac2aa8f35e0bc01","kind":"fingerprint_related","node_a_id":"n-5b6560967f09bc16","node_b_id":"n-24957611ccdbbb19","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-24957611ccdbbb19","other":{"id":"n-24957611ccdbbb19","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","720","324"],["8000","21600","19440","5832"],["160000","576000","777600","466560","104976"]],"card_rows":[["400","720","324"],["8000","21600","19440","5832"],["160000","576000","777600","466560","104976"],["3200000","14400000","25920000","23328000","10497600","1889568"]],"row_sums_preview":["1444","54872","2085136","79235168","3010936384","114415582592","4347792138496","165216101262848"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A007318","basis A013610"],"edges":["e-37fb8ca4839f71ce","e-65ceb6a2cd85f5d3"],"text":"A038284 \u2192 basis A007318 \u2192 basis A013610"},"headline":"A038284 \u2192 basis A007318 \u2192 basis A013610","chain":null}},{"id":"l-afbd8cc64fe2590d","kind":"fingerprint_related","node_a_id":"n-5b6560967f09bc16","node_b_id":"n-80e01f57edd0fcf9","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-80e01f57edd0fcf9","other":{"id":"n-80e01f57edd0fcf9","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","9"],["100","180","81"]],"card_rows":[["1"],["10","9"],["100","180","81"],["1000","2700","2430","729"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610","basis A013610"],"edges":["e-b9cdb58d54bdc029","e-0895947dc081c0b1"],"text":"A038255 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038255 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-e213573a63eca80d","kind":"fingerprint_related","node_a_id":"n-a76100adbc274ca7","node_b_id":"n-5b6560967f09bc16","fingerprint":"f6-249d425e0def4c4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-249d425e0def4c4f"},"other_id":"n-a76100adbc274ca7","other":{"id":"n-a76100adbc274ca7","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","first_index":0,"fingerprint":"f6-249d425e0def4c4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["18"],["720","648"]],"card_rows":[["18"],["720","648"],["21600","38880","17496"],["576000","1555200","1399680","419904"]],"row_sums_preview":["0","18","1368","77976","3950784","187662240","8557398144","379377984384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4e096b5929fd3f2e","steps":["basis A013609","D"],"edges":["e-66940ce826511afb","e-0cb5832302ddb9dc"],"text":"A038323 \u2192 basis A013609 \u2192 D"},"headline":"A038323 \u2192 basis A013609 \u2192 D","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:           18,            20;\n  2:          324,           720,           400;\n  3:         5832,         19440,         21600,           8000;\n  4:       104976,        466560,        777600,         576000,         160000;\n  5:      1889568,      10497600,      23328000,       25920000,       14400000,        3200000;\n  6:     34012224,     226748160,     629856000,      933120000,      777600000,      345600000,       64000000;\n  7:    612220032,    4761711360,   15872371200,    29393280000,    32659200000,    21772800000,     8064000000,     1280000000;\n  8:  11019960576,   97955205120,  380936908800,   846526464000,  1175731200000,  1045094400000,   580608000000,   184320000000,   25600000000;\n  9: 198359290368, 1983592903680, 8815968460800, 22856214528000, 38093690880000, 42326323200000, 31352832000000, 14929920000000, 4147200000000, 512000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+20*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[(18 + 20*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] = (18 + 20*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 A013609","basis A013609"],"edges":["e-c5caeaebd0a50686","e-510d3f765aca8c0c"],"text":"A013622 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A013622 \u2192 basis A013609 \u2192 basis A013609","chain":null}