{"id":"n-b3e0081a257e50f9","revision":0,"first_index":0,"fingerprint":"f6-27210632b0078bca","name":"basis_map_A013609 of basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^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"],["14","18"],["196","504","324"],["2744","10584","13608","5832"],["38416","197568","381024","326592","104976"],["537824","3457440","8890560","11430720","7348320","1889568"],["7529536","58084992","186701760","320060160","308629440","158723712","34012224"],["105413504","948721536","3659354496","7841473920","10081895040","7777461888","3333197952","612220032"],["1475789056","15179544576","68307950592","175649015808","282293061120","290358577152","186659085312","68568643584","11019960576"],["20661046784","239077827072","1229543110656","3688629331968","7113785140224","9146295180288","7839681583104","4319824545792","1388515032576","198359290368"],["289254654976","3718988421120","21517004436480","73772586639360","165988319938560","256096265048064","274388855408640","201591812136960","97196052280320","27770300651520","3570467226624"],["4049565169664","57272421685248","368179853690880","1420122292807680","3651743038648320","6573137469566976","8451176746586112","7761284767272960","4989397350389760","2138313150167040","549851952900096","64268410079232"],["56693912375296","874706076647424","6185421542006784","26508949465743360","76686603811614720","157755299269607424","236632948904411136","260779168180371456","209554688716369920","119745536409354240","46187564043608064","10797092893310976","1156831381426176"],["793714773254144","13266375495819264","102340610967748608","482462880276529152","1550773543745986560","3588933058383568896","6152456671514689536","7910301434804600832","7627790669275865088","5448421906625617920","2802045551978889216","982535453291298816","210543311419564032","20822964865671168"],["11112006825558016","200016122860044288","1671563312473227264","8596611321290883072","30395161457421336576","78158986604797722624","150735188452109893632","221488440174528823296","249174495196344926208","213578138739724222464","137300232046965571584","64192316281698189312","20633244519117275136","4081301113671548928","374813367582081024"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","32","1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832","1125899906842624","36028797018963968","1152921504606846976","36893488147419103232","1180591620717411303424"],"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","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200644,"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-90064c8a5828fc45"]},{"id":200645,"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-90064c8a5828fc45"]},{"id":200646,"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-90064c8a5828fc45"]}],"region_exact":{},"incoming_edges":[{"id":"e-90064c8a5828fc45","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-b3e0081a257e50f9","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":"53da69242cf03f1d426c2bd8b59484b69aa6cbc225c840fe918eb720db7e48c0","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-a13c2de84911142b","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^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-082ed6129c7fda24","kind":"fingerprint_related","node_a_id":"n-53dc6d1eeb1907f4","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-53dc6d1eeb1907f4","other":{"id":"n-53dc6d1eeb1907f4","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","504","196"],["5832","13608","10584","2744"],["104976","326592","381024","197568","38416"]],"card_rows":[["324","504","196"],["5832","13608","10584","2744"],["104976","326592","381024","197568","38416"],["1889568","7348320","11430720","8890560","3457440","537824"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013609"],"edges":["e-322a2e4c0bbd3231"],"text":"A038321 \u2192 basis A013609"},"headline":"A038321 \u2192 basis A013609","chain":null}},{"id":"l-1fee698df82caf39","kind":"fingerprint_related","node_a_id":"n-f29c7367d5544e24","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-f29c7367d5544e24","other":{"id":"n-f29c7367d5544e24","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["504","392"],["13608","21168","8232"],["326592","762048","592704","153664"]],"card_rows":[["504","392"],["13608","21168","8232"],["326592","762048","592704","153664"],["7348320","22861440","26671680","13829760","2689120"]],"row_sums_preview":["896","43008","1835008","73400320","2818572288","105226698752","3848290697216","138538465099776"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013609","D"],"edges":["e-322a2e4c0bbd3231","e-49464b0b3f306e05"],"text":"A038321 \u2192 basis A013609 \u2192 D"},"headline":"A038321 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-2793393ae7c33cff","kind":"fingerprint_related","node_a_id":"n-b3e0081a257e50f9","node_b_id":"n-2b84091cce44a505","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-2b84091cce44a505","other":{"id":"n-2b84091cce44a505","name":"basis_map_A013610 of basis_map_A007318 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","9"],["49","126","81"]],"card_rows":[["1"],["7","9"],["49","126","81"],["343","1323","1701","729"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A007318","basis A013610"],"edges":["e-5bade392cbb9f615","e-430cff3afedaf265"],"text":"A013610 \u2192 basis A007318 \u2192 basis A013610"},"headline":"A013610 \u2192 basis A007318 \u2192 basis A013610","chain":null}},{"id":"l-4bea8234121f194c","kind":"fingerprint_related","node_a_id":"n-91e60f9c0a45aa60","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-91e60f9c0a45aa60","other":{"id":"n-91e60f9c0a45aa60","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","504","324"],["2744","10584","13608","5832"],["38416","197568","381024","326592","104976"]],"card_rows":[["196","504","324"],["2744","10584","13608","5832"],["38416","197568","381024","326592","104976"],["537824","3457440","8890560","11430720","7348320","1889568"]],"row_sums_preview":["1024","32768","1048576","33554432","1073741824","34359738368","1099511627776","35184372088832"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013610"],"edges":["e-89b2e37c644537dd"],"text":"A038284 \u2192 basis A013610"},"headline":"A038284 \u2192 basis A013610","chain":null}},{"id":"l-6dadf99e89abefd3","kind":"fingerprint_related","node_a_id":"n-504bafdebe53d9d7","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-504bafdebe53d9d7","other":{"id":"n-504bafdebe53d9d7","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","27"],["441","1134","729"]],"card_rows":[["1"],["21","27"],["441","1134","729"],["9261","35721","45927","19683"]],"row_sums_preview":["1","48","2304","110592","5308416","254803968","12230590464","587068342272"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013610"],"edges":["e-b7b3e04e23eb02df"],"text":"A038335 \u2192 basis A013610"},"headline":"A038335 \u2192 basis A013610","chain":null}},{"id":"l-9208eaff10e86272","kind":"fingerprint_related","node_a_id":"n-b3e0081a257e50f9","node_b_id":"n-b0181b55e9572fd7","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-b0181b55e9572fd7","other":{"id":"n-b0181b55e9572fd7","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["784","2016","1296"],["21952","84672","108864","46656"],["614656","3161088","6096384","5225472","1679616"]],"card_rows":[["784","2016","1296"],["21952","84672","108864","46656"],["614656","3161088","6096384","5225472","1679616"],["17210368","110638080","284497920","365783040","235146240","60466176"]],"row_sums_preview":["4096","262144","16777216","1073741824","68719476736","4398046511104","281474976710656","18014398509481984"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013610","basis A013610"],"edges":["e-77aaa200c8a95097","e-9dfa1f1d35351bc0"],"text":"A038330 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038330 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-afcfb7e490e8d17b","kind":"fingerprint_related","node_a_id":"n-b80f414e96464f96","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-b80f414e96464f96","other":{"id":"n-b80f414e96464f96","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","9"],["49","126","81"]],"card_rows":[["1"],["7","9"],["49","126","81"],["343","1323","1701","729"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-4c464713c83c5f6b","e-7fd42eba8a8f2474"],"text":"A038255 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038255 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b79766da70318e28","kind":"fingerprint_related","node_a_id":"n-c39cdfaacf7f4a37","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-c39cdfaacf7f4a37","other":{"id":"n-c39cdfaacf7f4a37","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["27","21"],["729","1134","441"]],"card_rows":[["1"],["27","21"],["729","1134","441"],["19683","45927","35721","9261"]],"row_sums_preview":["1","48","2304","110592","5308416","254803968","12230590464","587068342272"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013610","reverse"],"edges":["e-b7b3e04e23eb02df","e-35c51cfc492cb26d"],"text":"A038335 \u2192 basis A013610 \u2192 reverse"},"headline":"A038335 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-c927f8934d9a63da","kind":"fingerprint_related","node_a_id":"n-7e7082631edb710e","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-7e7082631edb710e","other":{"id":"n-7e7082631edb710e","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["504","648"],["10584","27216","17496"],["197568","762048","979776","419904"]],"card_rows":[["504","648"],["10584","27216","17496"],["197568","762048","979776","419904"],["3457440","17781120","34292160","29393280","9447840"]],"row_sums_preview":["1152","55296","2359296","94371840","3623878656","135291469824","4947802324992","178120883699712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013610","D"],"edges":["e-89b2e37c644537dd","e-432da6ddf975cd40"],"text":"A038284 \u2192 basis A013610 \u2192 D"},"headline":"A038284 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-e583437a6349d960","kind":"fingerprint_related","node_a_id":"n-6678cccd35d698d6","node_b_id":"n-b3e0081a257e50f9","fingerprint":"f6-27210632b0078bca","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-27210632b0078bca"},"other_id":"n-6678cccd35d698d6","other":{"id":"n-6678cccd35d698d6","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-27210632b0078bca","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["27"],["1134","1458"]],"card_rows":[["27"],["1134","1458"],["35721","91854","59049"],["1000188","3857868","4960116","2125764"]],"row_sums_preview":["0","27","2592","186624","11943936","716636160","41278242816","2311581597696"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013610","D"],"edges":["e-b7b3e04e23eb02df","e-2f1a844045f80fe2"],"text":"A038335 \u2192 basis A013610 \u2192 D"},"headline":"A038335 \u2192 basis A013610 \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:          14,           18;\n  2:         196,          504,           324;\n  3:        2744,        10584,         13608,          5832;\n  4:       38416,       197568,        381024,        326592,        104976;\n  5:      537824,      3457440,       8890560,      11430720,       7348320,       1889568;\n  6:     7529536,     58084992,     186701760,     320060160,     308629440,     158723712,      34012224;\n  7:   105413504,    948721536,    3659354496,    7841473920,   10081895040,    7777461888,    3333197952,     612220032;\n  8:  1475789056,  15179544576,   68307950592,  175649015808,  282293061120,  290358577152,  186659085312,   68568643584,   11019960576;\n  9: 20661046784, 239077827072, 1229543110656, 3688629331968, 7113785140224, 9146295180288, 7839681583104, 4319824545792, 1388515032576, 198359290368;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+18*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 + 18*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 + 18*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-5733d79f45a14084","steps":["basis A013610","basis A013609"],"edges":["e-4eef4a156684ec8f","e-90064c8a5828fc45"],"text":"A013620 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A013620 \u2192 basis A013610 \u2192 basis A013609","chain":null}
