{"id":"n-67dfd62e07739d13","revision":1,"first_index":0,"fingerprint":"f6-5184fe74b2189306","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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"],["7","3"],["49","42","9"],["343","441","189","27"],["2401","4116","2646","756","81"],["16807","36015","30870","13230","2835","243"],["117649","302526","324135","185220","59535","10206","729"],["823543","2470629","3176523","2268945","972405","250047","35721","2187"],["5764801","19765032","29647548","25412184","13613670","4667544","1000188","122472","6561"],["40353607","155649627","266827932","266827932","171532242","73513818","21003948","3857868","413343","19683"],["282475249","1210608210","2334744405","2668279320","2001209490","1029193452","367569090","90016920","14467005","1377810","59049"],["1977326743","9321683217","19975035465","25682188455","22013304390","13207982634","5660563986","1732825710","371319795","53045685","4546773","177147"],["13841287201","71183762748","167790297906","239700425580","231139696095","158495791608","79247895804","29111471928","7797715695","1485279180","190964466","14880348","531441"],["96889010407","539810200839","1388083373586","2181273872778","2337079149405","1802889629541","1030222645452","441523990908","141918425649","33790101345","5792588802","677055834","48361131","1594323"],["678223072849","4069338437094","11336014217619","19433167230204","22903375664169","19631464855002","12620227406787","6181335872712","2318000952267","662285986362","141918425649","22117157244","2369695419","156243654","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","100","1000","10000","100000","1000000","10000000","100000000","1000000000","10000000000","100000000000","1000000000000","10000000000000","100000000000000"],"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":[["7","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167412,"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-b9cdb58d54bdc029"]},{"id":167413,"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-b9cdb58d54bdc029"]},{"id":167414,"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-b9cdb58d54bdc029"]}],"region_exact":{},"incoming_edges":[{"id":"e-b9cdb58d54bdc029","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-67dfd62e07739d13","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":"8035aa1fc2159a4d93b8e40b28cb92e4ae5bcd7649c35fc177a0486b7e260a80","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-22693abd431a71f2","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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-20299f9ab81f33e8","kind":"fingerprint_related","node_a_id":"n-6dcbb3626ca58345","node_b_id":"n-67dfd62e07739d13","fingerprint":"f6-5184fe74b2189306","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5184fe74b2189306"},"other_id":"n-6dcbb3626ca58345","other":{"id":"n-6dcbb3626ca58345","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-5184fe74b2189306","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","9"],["441","378","81"]],"card_rows":[["1"],["21","9"],["441","378","81"],["9261","11907","5103","729"]],"row_sums_preview":["1","30","900","27000","810000","24300000","729000000","21870000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A007318"],"edges":["e-9b0235f0075e4320"],"text":"A038335 \u2192 basis A007318"},"headline":"A038335 \u2192 basis A007318","chain":null}},{"id":"l-f2738164e6677ed1","kind":"fingerprint_related","node_a_id":"n-7288f584ad081a07","node_b_id":"n-67dfd62e07739d13","fingerprint":"f6-5184fe74b2189306","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5184fe74b2189306"},"other_id":"n-7288f584ad081a07","other":{"id":"n-7288f584ad081a07","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-5184fe74b2189306","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","168","36"],["2744","3528","1512","216"],["38416","65856","42336","12096","1296"]],"card_rows":[["196","168","36"],["2744","3528","1512","216"],["38416","65856","42336","12096","1296"],["537824","1152480","987840","423360","90720","7776"]],"row_sums_preview":["400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A007318"],"edges":["e-37fb8ca4839f71ce"],"text":"A038284 \u2192 basis A007318"},"headline":"A038284 \u2192 basis A007318","chain":null}},{"id":"l-1dd9927bebb06f64","kind":"oeis_match","node_a_id":"n-67dfd62e07739d13","node_b_id":null,"fingerprint":null,"oeis_id":"A155148","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A155148","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-643c060a772bd5e1","kind":"oeis_match","node_a_id":"n-67dfd62e07739d13","node_b_id":null,"fingerprint":null,"oeis_id":"A038269","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":37},"in_catalog":false,"oeis_id":"A038269","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6bec1455446fc23b","kind":"oeis_match","node_a_id":"n-67dfd62e07739d13","node_b_id":null,"fingerprint":null,"oeis_id":"A178500","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A178500","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6cff5a9aab7e284b","kind":"oeis_match","node_a_id":"n-67dfd62e07739d13","node_b_id":null,"fingerprint":null,"oeis_id":"A013624","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"reversed_triangle","terms_compared":38},"in_catalog":false,"oeis_id":"A013624","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9de0a8c4e224e6d0","kind":"oeis_match","node_a_id":"n-67dfd62e07739d13","node_b_id":null,"fingerprint":null,"oeis_id":"A011557","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A011557","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ec09101c23ccaef0","kind":"oeis_match","node_a_id":"n-67dfd62e07739d13","node_b_id":null,"fingerprint":null,"oeis_id":"A178501","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A178501","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A013624","relation":"match_reversed_triangle"},{"oeis_id":"A011557","relation":"match_row_sums"},{"oeis_id":"A155148","relation":"match_row_sums"},{"oeis_id":"A178500","relation":"match_row_sums"},{"oeis_id":"A178501","relation":"match_row_sums"},{"oeis_id":"A038269","relation":"match_triangle"}],"bfiles":[{"node_id":"n-67dfd62e07739d13","oeis_id":"A011557","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":20990},{"node_id":"n-67dfd62e07739d13","oeis_id":"A013624","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":38,"oeis_terms_agreeing":38,"oeis_terms_available":38,"term_count":990,"raw_size":29421},{"node_id":"n-67dfd62e07739d13","oeis_id":"A038269","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":37,"oeis_terms_agreeing":37,"oeis_terms_available":37,"term_count":990,"raw_size":29421}],"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:        7,         3;\n  2:       49,        42,         9;\n  3:      343,       441,       189,        27;\n  4:     2401,      4116,      2646,       756,        81;\n  5:    16807,     36015,     30870,     13230,      2835,      243;\n  6:   117649,    302526,    324135,    185220,     59535,    10206,      729;\n  7:   823543,   2470629,   3176523,   2268945,    972405,   250047,    35721,    2187;\n  8:  5764801,  19765032,  29647548,  25412184,  13613670,  4667544,  1000188,  122472,   6561;\n  9: 40353607, 155649627, 266827932, 266827932, 171532242, 73513818, 21003948, 3857868, 413343, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+3*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[(7 + 3*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] = (7 + 3*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}},{"oeis_id":"A013624","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3,      7;\n  2:     9,     42,      49;\n  3:    27,    189,     441,      343;\n  4:    81,    756,    2646,     4116,     2401;\n  5:   243,   2835,   13230,    30870,    36015,     16807;\n  6:   729,  10206,   59535,   185220,   324135,    302526,    117649;\n  7:  2187,  35721,  250047,   972405,  2268945,   3176523,   2470629,    823543;\n  8:  6561, 122472, 1000188,  4667544, 13613670,  25412184,  29647548,  19765032,   5764801;\n  9: 19683, 413343, 3857868, 21003948, 73513818, 171532242, 266827932, 266827932, 155649627, 40353607;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+3*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[(7 + 3*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (7 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A038269","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        7,         3;\n  2:       49,        42,         9;\n  3:      343,       441,       189,        27;\n  4:     2401,      4116,      2646,       756,        81;\n  5:    16807,     36015,     30870,     13230,      2835,      243;\n  6:   117649,    302526,    324135,    185220,     59535,    10206,      729;\n  7:   823543,   2470629,   3176523,   2268945,    972405,   250047,    35721,    2187;\n  8:  5764801,  19765032,  29647548,  25412184,  13613670,  4667544,  1000188,  122472,   6561;\n  9: 40353607, 155649627, 266827932, 266827932, 171532242, 73513818, 21003948, 3857868, 413343, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+3*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[(7 + 3*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (7 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610"],"edges":["e-b9cdb58d54bdc029"],"text":"A038255 \u2192 basis A013610"},"headline":"A038255 \u2192 basis A013610","chain":null}
