{"id":"n-03dee8d451c5f864","revision":0,"first_index":0,"fingerprint":"f6-5184fe74b2189306","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^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"],["14","6"],["196","168","36"],["2744","3528","1512","216"],["38416","65856","42336","12096","1296"],["537824","1152480","987840","423360","90720","7776"],["7529536","19361664","20744640","11854080","3810240","653184","46656"],["105413504","316240512","406594944","290424960","124467840","32006016","4572288","279936"],["1475789056","5059848192","7589772288","6505519104","3485099520","1194891264","256048128","31352832","1679616"],["20661046784","79692609024","136615901184","136615901184","87824507904","37639074816","10754021376","1975228416","211631616","10077696"],["289254654976","1239662807040","2390778270720","2732318023680","2049238517760","1053894094848","376390748160","92177326080","14814213120","1410877440","60466176"],["4049565169664","19090807228416","40908872632320","52597121955840","45083247390720","27049948434432","11592835043328","3548827054080","760462940160","108637562880","9311791104","362797056"],["56693912375296","291568692215808","687269060222976","981812943175680","946748195205120","649198762426368","324599381213184","119240589017088","31939443486720","6083703521280","782190452736","60949905408","2176782336"],["793714773254144","4422125165273088","11371178996416512","17868995565797376","19145352391925760","14769271845199872","8439583911542784","3616964533518336","1162595742916608","276808510218240","47452887465984","5546441392128","396174385152","13060694016"],["11112006825558016","66672040953348096","185729256941469696","318393011899662336","375248906881744896","321641920184352768","206769805832798208","101275006938513408","37978127601942528","10850893600555008","2325191485833216","362367504285696","38825089744896","2559896027136","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","20","400","8000","160000","3200000","64000000","1280000000","25600000000","512000000000","10240000000000","204800000000000","4096000000000000","81920000000000000","1638400000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["14","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200634,"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-a5489276a0ebcf7b"]},{"id":200635,"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-a5489276a0ebcf7b"]},{"id":200636,"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-a5489276a0ebcf7b"]}],"region_exact":{},"incoming_edges":[{"id":"e-a5489276a0ebcf7b","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-03dee8d451c5f864","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":"58b95521a8e199f92b3c6fab4f00c45cb83c36727e21cb87165d15ae3384c679","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-3c11bfaa4e68c9bc","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^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-2087e63ed58ec1ef","kind":"fingerprint_related","node_a_id":"n-7c877ed9ba2e6414","node_b_id":"n-03dee8d451c5f864","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-7c877ed9ba2e6414","other":{"id":"n-7c877ed9ba2e6414","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","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":[["3"],["9","21"],["27","126","147"]],"card_rows":[["3"],["9","21"],["27","126","147"],["81","567","1323","1029"]],"row_sums_preview":["3","30","300","3000","30000","300000","3000000","30000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-1a05d30902084644","e-e3646c640befb9d9"],"text":"A120910 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A120910 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3833cde9796928cc","kind":"fingerprint_related","node_a_id":"n-67dfd62e07739d13","node_b_id":"n-03dee8d451c5f864","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-67dfd62e07739d13","other":{"id":"n-67dfd62e07739d13","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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"],["7","3"],["49","42","9"]],"card_rows":[["1"],["7","3"],["49","42","9"],["343","441","189","27"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610"],"edges":["e-b9cdb58d54bdc029"],"text":"A038255 \u2192 basis A013610"},"headline":"A038255 \u2192 basis A013610","chain":null}},{"id":"l-4f397aff2bd27515","kind":"fingerprint_related","node_a_id":"n-94d6274c5f252185","node_b_id":"n-03dee8d451c5f864","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-94d6274c5f252185","other":{"id":"n-94d6274c5f252185","name":"reversal of 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":[["36","168","196"],["216","1512","3528","2744"],["1296","12096","42336","65856","38416"]],"card_rows":[["36","168","196"],["216","1512","3528","2744"],["1296","12096","42336","65856","38416"],["7776","90720","423360","987840","1152480","537824"]],"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","reverse"],"edges":["e-37fb8ca4839f71ce","e-57e373e08b265a38"],"text":"A038284 \u2192 basis A007318 \u2192 reverse"},"headline":"A038284 \u2192 basis A007318 \u2192 reverse","chain":null}},{"id":"l-55fb0bb25cb35566","kind":"fingerprint_related","node_a_id":"n-7288f584ad081a07","node_b_id":"n-03dee8d451c5f864","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]","(-inf, -1]"],"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-63e272c0d1739e86","kind":"fingerprint_related","node_a_id":"n-3fcc1ea45b099eca","node_b_id":"n-03dee8d451c5f864","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-3fcc1ea45b099eca","other":{"id":"n-3fcc1ea45b099eca","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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":[[],["3"],["42","18"]],"card_rows":[["3"],["42","18"],["441","378","81"],["4116","5292","2268","324"]],"row_sums_preview":["0","3","60","900","12000","150000","1800000","21000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610","D"],"edges":["e-b9cdb58d54bdc029","e-108da488069017c7"],"text":"A038255 \u2192 basis A013610 \u2192 D"},"headline":"A038255 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-72e94874cb4a35b8","kind":"fingerprint_related","node_a_id":"n-846c10917dd517bf","node_b_id":"n-03dee8d451c5f864","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-846c10917dd517bf","other":{"id":"n-846c10917dd517bf","name":"reversal of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-5184fe74b2189306","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","21"],["81","378","441"]],"card_rows":[["1"],["9","21"],["81","378","441"],["729","5103","11907","9261"]],"row_sums_preview":["1","30","900","27000","810000","24300000","729000000","21870000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1","reverse"],"edges":["e-62ade03cd9f25f04","e-4b512ae2958817e5"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-ae526e1c0b4f6762","kind":"fingerprint_related","node_a_id":"n-746d6a84958f28b6","node_b_id":"n-03dee8d451c5f864","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-746d6a84958f28b6","other":{"id":"n-746d6a84958f28b6","name":"derivative of 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":[["168","72"],["3528","3024","648"],["65856","84672","36288","5184"]],"card_rows":[["168","72"],["3528","3024","648"],["65856","84672","36288","5184"],["1152480","1975680","1270080","362880","38880"]],"row_sums_preview":["240","7200","192000","4800000","115200000","2688000000","61440000000","1382400000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A007318","D"],"edges":["e-37fb8ca4839f71ce","e-05310e1f064273fd"],"text":"A038284 \u2192 basis A007318 \u2192 D"},"headline":"A038284 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-b2a16f63c561702f","kind":"fingerprint_related","node_a_id":"n-6dcbb3626ca58345","node_b_id":"n-03dee8d451c5f864","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]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1"],"edges":["e-62ade03cd9f25f04"],"text":"A038335 \u2192 x \u21a6 x+1"},"headline":"A038335 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-b74e9e6d1a04b5c7","kind":"fingerprint_related","node_a_id":"n-cb48487a82c52afe","node_b_id":"n-03dee8d451c5f864","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-cb48487a82c52afe","other":{"id":"n-cb48487a82c52afe","name":"derivative of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-5184fe74b2189306","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["9"],["378","162"]],"card_rows":[["9"],["378","162"],["11907","10206","2187"],["333396","428652","183708","26244"]],"row_sums_preview":["0","9","540","24300","972000","36450000","1312200000","45927000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["x \u21a6 x+1","D"],"edges":["e-62ade03cd9f25f04","e-5fe52ba8cb7bc1c7"],"text":"A038335 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A038335 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-be28b6f970f8a5b1","kind":"fingerprint_related","node_a_id":"n-ab158bcd9cd8fa53","node_b_id":"n-03dee8d451c5f864","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-ab158bcd9cd8fa53","other":{"id":"n-ab158bcd9cd8fa53","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-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"],["3","7"],["9","42","49"]],"card_rows":[["1"],["3","7"],["9","42","49"],["27","189","441","343"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610","reverse"],"edges":["e-b9cdb58d54bdc029","e-e3dea0c2f6b14889"],"text":"A038255 \u2192 basis A013610 \u2192 reverse"},"headline":"A038255 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-c122819e1379bca1","kind":"fingerprint_related","node_a_id":"n-ca0022e1bdb371d2","node_b_id":"n-03dee8d451c5f864","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-ca0022e1bdb371d2","other":{"id":"n-ca0022e1bdb371d2","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (1+2*x)^n.","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"],["3","7"],["9","42","49"]],"card_rows":[["1"],["3","7"],["9","42","49"],["27","189","441","343"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-95924f9cf72f1e84","e-1f8e28813e5553dc"],"text":"A013609 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A013609 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-eb2bc082114b12c6","kind":"fingerprint_related","node_a_id":"n-fcc243fd8c985b80","node_b_id":"n-03dee8d451c5f864","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-fcc243fd8c985b80","other":{"id":"n-fcc243fd8c985b80","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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":[["784","672","144"],["21952","28224","12096","1728"],["614656","1053696","677376","193536","20736"]],"card_rows":[["784","672","144"],["21952","28224","12096","1728"],["614656","1053696","677376","193536","20736"],["17210368","36879360","31610880","13547520","2903040","248832"]],"row_sums_preview":["1600","64000","2560000","102400000","4096000000","163840000000","6553600000000","262144000000000"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013610","basis A007318"],"edges":["e-77aaa200c8a95097","e-5a6fb2c20be4892d"],"text":"A038330 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038330 \u2192 basis A013610 \u2192 basis A007318","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:          14,           6;\n  2:         196,         168,           36;\n  3:        2744,        3528,         1512,          216;\n  4:       38416,       65856,        42336,        12096,        1296;\n  5:      537824,     1152480,       987840,       423360,       90720,        7776;\n  6:     7529536,    19361664,     20744640,     11854080,     3810240,      653184,       46656;\n  7:   105413504,   316240512,    406594944,    290424960,   124467840,    32006016,     4572288,     279936;\n  8:  1475789056,  5059848192,   7589772288,   6505519104,  3485099520,  1194891264,   256048128,   31352832,   1679616;\n  9: 20661046784, 79692609024, 136615901184, 136615901184, 87824507904, 37639074816, 10754021376, 1975228416, 211631616, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+6*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[(14 + 6*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] = (14 + 6*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-a920b56c9b3ef4d1","steps":["basis A013610","basis A013609"],"edges":["e-254f1b828fe6620e","e-a5489276a0ebcf7b"],"text":"A038303 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038303 \u2192 basis A013610 \u2192 basis A013609","chain":null}