{"id":"n-3992e0c998ec014d","revision":0,"first_index":0,"fingerprint":"f6-da2debf48cb6165b","name":"basis_map_A013609 of basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","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"],["10","12"],["150","360","216"],["3000","10800","12960","5184"],["75000","360000","648000","518400","155520"],["2250000","13500000","32400000","38880000","23328000","5598720"],["78750000","567000000","1701000000","2721600000","2449440000","1175731200","235146240"],["3150000000","26460000000","95256000000","190512000000","228614400000","164602368000","65840947200","11287019520"],["141750000000","1360800000000","5715360000000","13716864000000","20575296000000","19752284160000","11851370496000","4063327027200","609499054080"],["7087500000000","76545000000000","367416000000000","1028764800000000","1851776640000000","2222131968000000","1777705574400000","914248581120000","274274574336000","36569943244800"],["389812500000000","4677750000000000","25259850000000000","80831520000000000","169746192000000000","244434516480000000","244434516480000000","167612239872000000","75425507942400000","20113468784640000","2413616254156800"],["23388750000000000","308731500000000000","1852389000000000000","6668600400000000000","16004640960000000000","26887796812800000000","32265356175360000000","27656019578880000000","16593611747328000000","6637444698931200000","1592986727743488000","173780370299289600"],["1520268750000000000","21891870000000000000","144486342000000000000","577945368000000000000","1560452493600000000000","2996068787712000000000","4194496302796800000000","4314339054305280000000","3235754290728960000000","1725735621722112000000","621264823819960320000","135548688833445888000","13554868883344588800"],["106418812500000000000","1660133475000000000000","11952961020000000000000","52593028488000000000000","157779085464000000000000","340802824602240000000000","545284519363584000000000","654341423236300800000000","588907280912670720000000","392604853941780480000000","188450329892054630400000","61674653419217879040000","12334930683843575808000","1138608986200945459200"],["7981410937500000000000","134087703750000000000000","1045884089250000000000000","5020243628400000000000000","16566803973720000000000000","39760329536928000000000000","71568593166470400000000000","98151213485445120000000000","103058774159717376000000000","82447019327773900800000000","49468211596664340480000000","21586128696726257664000000","6475838609017877299200000","1195539435510992732160000","102474808758085091328000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","22","726","31944","1756920","115956720","8928667440","785722734720","77786550737280","8556520581100800","1035338990313196800","136664746721341977600","19543058781151902796800","3009631052297393030707200","496589123629069850066688000"],"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":[["5","6"],["5","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200913,"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-7b1159fca4765fad"]},{"id":200914,"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-7b1159fca4765fad"]},{"id":200915,"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-7b1159fca4765fad"]},{"id":200916,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-7b1159fca4765fad"]}],"region_exact":{},"incoming_edges":[{"id":"e-7b1159fca4765fad","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-3992e0c998ec014d","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":"f3adee6828f89137985ea75bf9c051451bb67491e1e0c6719c7d789b9b3b3643","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9c0ed28aed848f75","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","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-03791a282b105229","kind":"fingerprint_related","node_a_id":"n-203a9741970ef52c","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-203a9741970ef52c","other":{"id":"n-203a9741970ef52c","name":"derivative of basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["60","50"],["540","900","375"],["4320","10800","9000","2500"]],"card_rows":[["60","50"],["540","900","375"],["4320","10800","9000","2500"],["32400","108000","135000","75000","15625"]],"row_sums_preview":["110","1815","26620","366025","4831530","62004635","779486840","9646149645"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x+1","D"],"edges":["e-7ae18e4d322020fd","e-46969d268abb8e5d"],"text":"A013612 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A013612 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-0c50d7f1166da0f5","kind":"fingerprint_related","node_a_id":"n-e6ebe74518d42cbe","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-e6ebe74518d42cbe","other":{"id":"n-e6ebe74518d42cbe","name":"Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","5"],["36","60","25"]],"card_rows":[["1"],["6","5"],["36","60","25"],["216","540","450","125"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038259"],"derivation":{"root":"n-e6ebe74518d42cbe","steps":[],"edges":[],"text":"A038259"},"headline":"A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","chain":null}},{"id":"l-12616c6fd980c45a","kind":"fingerprint_related","node_a_id":"n-cc913eb401ac41a8","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-cc913eb401ac41a8","other":{"id":"n-cc913eb401ac41a8","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["240","288"],["3600","8640","5184"],["48000","172800","207360","82944"]],"card_rows":[["240","288"],["3600","8640","5184"],["48000","172800","207360","82944"],["600000","2880000","5184000","4147200","1244160"]],"row_sums_preview":["528","17424","511104","14055360","371061504","9523911936","239458357248","5926594341888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["D"],"edges":["e-af6c7ab7c6383873"],"text":"A038314 \u2192 D"},"headline":"A038314 \u2192 D","chain":null}},{"id":"l-1b346e12afcb2a25","kind":"fingerprint_related","node_a_id":"n-2cf8ae9156c81000","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-2cf8ae9156c81000","other":{"id":"n-2cf8ae9156c81000","name":"basis_map_A013609 of basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","6"],["50","120","72"]],"card_rows":[["1"],["5","6"],["50","120","72"],["750","2700","3240","1296"]],"row_sums_preview":["1","11","242","7986","351384","19326120","1275523920","98215341840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610","basis A013609"],"edges":["e-542183e09dec6678","e-6508c17391606fa6"],"text":"A196347 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A196347 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-1ea1daa5145e9e44","kind":"fingerprint_related","node_a_id":"n-4550174e70f7b7f7","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-4550174e70f7b7f7","other":{"id":"n-4550174e70f7b7f7","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","60","25"],["216","540","450","125"],["1296","4320","5400","3000","625"]],"card_rows":[["36","60","25"],["216","540","450","125"],["1296","4320","5400","3000","625"],["7776","32400","54000","45000","18750","3125"]],"row_sums_preview":["121","1331","14641","161051","1771561","19487171","214358881","2357947691"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x+1"],"edges":["e-7ae18e4d322020fd"],"text":"A013612 \u2192 x \u21a6 x+1"},"headline":"A013612 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-214141b5a84d374b","kind":"fingerprint_related","node_a_id":"n-3a19ecc74ecdac2d","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-3a19ecc74ecdac2d","other":{"id":"n-3a19ecc74ecdac2d","name":"basis_map_A013609 of basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["10","12"],["50","120","72"]],"card_rows":[["2"],["10","12"],["50","120","72"],["250","900","1080","432"]],"row_sums_preview":["2","22","242","2662","29282","322102","3543122","38974342"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610","basis A013609"],"edges":["e-28b777b3c0b25332","e-5ab253fe7c88fc30"],"text":"A028326 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A028326 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-289135fb09591716","kind":"fingerprint_related","node_a_id":"n-1f0275834dfc668d","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-1f0275834dfc668d","other":{"id":"n-1f0275834dfc668d","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["5"],["60","50"]],"card_rows":[["5"],["60","50"],["540","900","375"],["4320","10800","9000","2500"]],"row_sums_preview":["0","5","110","1815","26620","366025","4831530","62004635"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["D"],"edges":["e-e3b76db75beab297"],"text":"A038259 \u2192 D"},"headline":"A038259 \u2192 D","chain":null}},{"id":"l-2bfa434e81ad3d1f","kind":"fingerprint_related","node_a_id":"n-30ad089f8e44b93b","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-30ad089f8e44b93b","other":{"id":"n-30ad089f8e44b93b","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","12"],["100","240","144"]],"card_rows":[["1"],["10","12"],["100","240","144"],["1000","3600","4320","1728"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013610","basis A013611"],"edges":["e-b9cdb58d54bdc029","e-4d3e879d738dcc0b"],"text":"A038255 \u2192 basis A013610 \u2192 basis A013611"},"headline":"A038255 \u2192 basis A013610 \u2192 basis A013611","chain":null}},{"id":"l-4c1fb05d9d9b13f4","kind":"fingerprint_related","node_a_id":"n-4739a5ec94dc3fc8","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-4739a5ec94dc3fc8","other":{"id":"n-4739a5ec94dc3fc8","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","18"],["225","540","324"]],"card_rows":[["1"],["15","18"],["225","540","324"],["3375","12150","14580","5832"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013610","basis A013609"],"edges":["e-b88e4ed4246b1dea","e-68ce35cc4e745ec7"],"text":"A038221 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038221 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-4eea9a20dfade55b","kind":"fingerprint_related","node_a_id":"n-4db9b5235c5a3916","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-4db9b5235c5a3916","other":{"id":"n-4db9b5235c5a3916","name":"derivative of basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["60","72"]],"card_rows":[["6"],["60","72"],["450","1080","648"],["3000","10800","12960","5184"]],"row_sums_preview":["0","6","132","2178","31944","439230","5797836","74405562"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013610","D"],"edges":["e-32b2129ae50cd22c","e-e42ff0de83917523"],"text":"A038220 \u2192 basis A013610 \u2192 D"},"headline":"A038220 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-4f9d2dcc800efae9","kind":"fingerprint_related","node_a_id":"n-9a00fc9238e55143","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-9a00fc9238e55143","other":{"id":"n-9a00fc9238e55143","name":"basis_map_A013609 of basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","6"],["50","120","72"]],"card_rows":[["1"],["5","6"],["50","120","72"],["625","2250","2700","1080"]],"row_sums_preview":["1","11","242","6655","204974","6764142","233846052","8359996359"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610","basis A013609"],"edges":["e-13e67b7b83b67a66","e-6268bc5e53877a5b"],"text":"A085880 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A085880 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-53a149f456dfc93b","kind":"fingerprint_related","node_a_id":"n-3992e0c998ec014d","node_b_id":"n-7a15286cc13dd028","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-7a15286cc13dd028","other":{"id":"n-7a15286cc13dd028","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3600","8640","5184"],["216000","777600","933120","373248"],["12960000","62208000","111974400","89579520","26873856"]],"card_rows":[["3600","8640","5184"],["216000","777600","933120","373248"],["12960000","62208000","111974400","89579520","26873856"],["777600000","4665600000","11197440000","13436928000","8062156800","1934917632"]],"row_sums_preview":["17424","2299968","303595776","40074642432","5289852801024","698260569735168","92170395205042176","12166492167065567232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013610","basis A013609"],"edges":["e-85e8843cd1dca083","e-bf32e7cd714922c6"],"text":"A038338 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038338 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-5bf753b36c2294aa","kind":"fingerprint_related","node_a_id":"n-31672dcf78468367","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-31672dcf78468367","other":{"id":"n-31672dcf78468367","name":"basis_map_A013609 of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["25","30"],["375","900","540"]],"card_rows":[["1"],["25","30"],["375","900","540"],["4375","15750","18900","7560"]],"row_sums_preview":["1","55","1815","46585","1024870","20292426","372027810","6430766430"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610","basis A013609"],"edges":["e-0c010b1c5e20b0cb","e-22c6f3190cf7b291"],"text":"A178821 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A178821 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-5fbb3a5238109c14","kind":"fingerprint_related","node_a_id":"n-e5535bb2038f7b5a","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-e5535bb2038f7b5a","other":{"id":"n-e5535bb2038f7b5a","name":"basis_map_A013609 of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","24"],["250","600","360"]],"card_rows":[["1"],["20","24"],["250","600","360"],["2500","9000","10800","4320"]],"row_sums_preview":["1","44","1210","26620","512435","9018856","148811124","2338460520"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013610","basis A013609"],"edges":["e-24fdbe8219e7135e","e-53fc9f0f71c94e01"],"text":"A178820 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A178820 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-69a4447dae08f47d","kind":"fingerprint_related","node_a_id":"n-9753d9ebb06d58a4","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-9753d9ebb06d58a4","other":{"id":"n-9753d9ebb06d58a4","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["10"],["240","200"]],"card_rows":[["10"],["240","200"],["4320","7200","3000"],["69120","172800","144000","40000"]],"row_sums_preview":["0","10","440","14520","425920","11712800","309217920","7936593280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013609","D"],"edges":["e-e169cb33ba7b2c8c","e-c7c0102504cc799c"],"text":"A038271 \u2192 basis A013609 \u2192 D"},"headline":"A038271 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-6d43fee9e56dc7bc","kind":"fingerprint_related","node_a_id":"n-d3da9e78959a3b34","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-d3da9e78959a3b34","other":{"id":"n-d3da9e78959a3b34","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","540","324"],["3375","12150","14580","5832"],["50625","243000","437400","349920","104976"]],"card_rows":[["225","540","324"],["3375","12150","14580","5832"],["50625","243000","437400","349920","104976"],["759375","4556250","10935000","13122000","7873200","1889568"]],"row_sums_preview":["1089","35937","1185921","39135393","1291467969","42618442977","1406408618241","46411484401953"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-5083e5230277d126","e-5373d706842ddf1c"],"text":"A038329 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A038329 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-86d5941eb325eb19","kind":"fingerprint_related","node_a_id":"n-3992e0c998ec014d","node_b_id":"n-d5a40cb29a976c68","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-d5a40cb29a976c68","other":{"id":"n-d5a40cb29a976c68","name":"basis_map_A013610 of basis_map_A007318 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-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","18"],["75","180","108"]],"card_rows":[["3"],["15","18"],["75","180","108"],["375","1350","1620","648"]],"row_sums_preview":["3","33","363","3993","43923","483153","5314683","58461513"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013609","basis A013610"],"edges":["e-618e7f7547136f14","e-e20b835f0843f8b6"],"text":"A120910 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A120910 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-8c2828c226082ef6","kind":"fingerprint_related","node_a_id":"n-190c5af041de2688","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-190c5af041de2688","other":{"id":"n-190c5af041de2688","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["240","200"],["4320","7200","3000"],["69120","172800","144000","40000"]],"card_rows":[["240","200"],["4320","7200","3000"],["69120","172800","144000","40000"],["1036800","3456000","4320000","2400000","500000"]],"row_sums_preview":["440","14520","425920","11712800","309217920","7936593280","199548631040","4938828618240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":["D"],"edges":["e-09922198ce5bd357"],"text":"A038336 \u2192 D"},"headline":"A038336 \u2192 D","chain":null}},{"id":"l-9069e08da1c7490b","kind":"fingerprint_related","node_a_id":"n-ec89fb3fc20684fc","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-ec89fb3fc20684fc","other":{"id":"n-ec89fb3fc20684fc","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","240","100"],["1728","4320","3600","1000"],["20736","69120","86400","48000","10000"]],"card_rows":[["144","240","100"],["1728","4320","3600","1000"],["20736","69120","86400","48000","10000"],["248832","1036800","1728000","1440000","600000","100000"]],"row_sums_preview":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec89fb3fc20684fc","steps":[],"edges":[],"text":"A038336"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.","chain":null}},{"id":"l-be05e91c8c680f50","kind":"fingerprint_related","node_a_id":"n-4f44de8e265280a7","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-4f44de8e265280a7","other":{"id":"n-4f44de8e265280a7","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","10"],["144","240","100"]],"card_rows":[["1"],["12","10"],["144","240","100"],["1728","4320","3600","1000"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013609"],"edges":["e-e169cb33ba7b2c8c"],"text":"A038271 \u2192 basis A013609"},"headline":"A038271 \u2192 basis A013609","chain":null}},{"id":"l-bea35360d43c4cc6","kind":"fingerprint_related","node_a_id":"n-8d841178f57a6d90","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-8d841178f57a6d90","other":{"id":"n-8d841178f57a6d90","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (14 + x)^n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","18"],["225","540","324"]],"card_rows":[["1"],["15","18"],["225","540","324"],["3375","12150","14580","5832"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a679863398df5985","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-011a72a323f5e83e","e-bfcb52c5c5266985"],"text":"A147716 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A147716 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-d0780d9bab52f78f","kind":"fingerprint_related","node_a_id":"n-6081e8199e2268e3","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-6081e8199e2268e3","other":{"id":"n-6081e8199e2268e3","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","960","576"],["8000","28800","34560","13824"],["160000","768000","1382400","1105920","331776"]],"card_rows":[["400","960","576"],["8000","28800","34560","13824"],["160000","768000","1382400","1105920","331776"],["3200000","19200000","46080000","55296000","33177600","7962624"]],"row_sums_preview":["1936","85184","3748096","164916224","7256313856","319277809664","14048223625216","618121839509504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013610"],"edges":["e-889e30cc3b49f76a"],"text":"A038334 \u2192 basis A013610"},"headline":"A038334 \u2192 basis A013610","chain":null}},{"id":"l-d092aa35c2e7ac30","kind":"fingerprint_related","node_a_id":"n-e3ea08b0c528c3d7","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-e3ea08b0c528c3d7","other":{"id":"n-e3ea08b0c528c3d7","name":"basis_map_A013609 of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","36"],["525","1260","756"]],"card_rows":[["1"],["30","36"],["525","1260","756"],["7000","25200","30240","12096"]],"row_sums_preview":["1","66","2541","74536","1844766","40584852","818461182","15433839432"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610","basis A013609"],"edges":["e-3fa86ece9d0bedd1","e-2868b2cd7321660d"],"text":"A178822 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A178822 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-d0d16aab07342d04","kind":"fingerprint_related","node_a_id":"n-29d7c5a1e19b9d74","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-29d7c5a1e19b9d74","other":{"id":"n-29d7c5a1e19b9d74","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["100","240","144"],["1000","3600","4320","1728"],["10000","48000","86400","69120","20736"]],"card_rows":[["100","240","144"],["1000","3600","4320","1728"],["10000","48000","86400","69120","20736"],["100000","600000","1440000","1728000","1036800","248832"]],"row_sums_preview":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":[],"edges":[],"text":"A038314"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","chain":null}},{"id":"l-d62a90390da86adf","kind":"fingerprint_related","node_a_id":"n-aa3a32290f637d6d","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-aa3a32290f637d6d","other":{"id":"n-aa3a32290f637d6d","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","12"],["100","240","144"]],"card_rows":[["1"],["10","12"],["100","240","144"],["1000","3600","4320","1728"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-88ca34e9877f4f47","e-98047256d05026af"],"text":"A038291 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038291 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-e8d4e966c0309a5c","kind":"fingerprint_related","node_a_id":"n-5e697901ced65a58","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-5e697901ced65a58","other":{"id":"n-5e697901ced65a58","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["960","1152"],["28800","69120","41472"],["768000","2764800","3317760","1327104"]],"card_rows":[["960","1152"],["28800","69120","41472"],["768000","2764800","3317760","1327104"],["19200000","92160000","165888000","132710400","39813120"]],"row_sums_preview":["2112","139392","8177664","449771520","23747936256","1219060727808","61301339455488","3034416303046656"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013610","D"],"edges":["e-889e30cc3b49f76a","e-56933e04628fa83f"],"text":"A038334 \u2192 basis A013610 \u2192 D"},"headline":"A038334 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-f05b42e340c16c79","kind":"fingerprint_related","node_a_id":"n-740344b12e5a271f","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-740344b12e5a271f","other":{"id":"n-740344b12e5a271f","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","960","400"],["13824","34560","28800","8000"],["331776","1105920","1382400","768000","160000"]],"card_rows":[["576","960","400"],["13824","34560","28800","8000"],["331776","1105920","1382400","768000","160000"],["7962624","33177600","55296000","46080000","19200000","3200000"]],"row_sums_preview":["1936","85184","3748096","164916224","7256313856","319277809664","14048223625216","618121839509504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013610","reverse"],"edges":["e-889e30cc3b49f76a","e-276e85e2869ef235"],"text":"A038334 \u2192 basis A013610 \u2192 reverse"},"headline":"A038334 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-f7a38cca1a795e75","kind":"fingerprint_related","node_a_id":"n-3addd939832b8a3b","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-3addd939832b8a3b","other":{"id":"n-3addd939832b8a3b","name":"reversal of basis_map_A007318 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["25","60","36"],["125","450","540","216"],["625","3000","5400","4320","1296"]],"card_rows":[["25","60","36"],["125","450","540","216"],["625","3000","5400","4320","1296"],["3125","18750","45000","54000","32400","7776"]],"row_sums_preview":["121","1331","14641","161051","1771561","19487171","214358881","2357947691"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["x \u21a6 x+1","reverse"],"edges":["e-7ae18e4d322020fd","e-61c2e6eb683b69e1"],"text":"A013612 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A013612 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-f9a8c8f2e760fde0","kind":"fingerprint_related","node_a_id":"n-c1936dc488503f26","node_b_id":"n-3992e0c998ec014d","fingerprint":"f6-da2debf48cb6165b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-da2debf48cb6165b"},"other_id":"n-c1936dc488503f26","other":{"id":"n-c1936dc488503f26","name":"basis_map_A013609 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-da2debf48cb6165b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","6"],["25","60","36"]],"card_rows":[["1"],["5","6"],["25","60","36"],["125","450","540","216"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-01cea33b6485d9cc","steps":["basis A013610"],"edges":["e-32b2129ae50cd22c"],"text":"A038220 \u2192 basis A013610"},"headline":"A038220 \u2192 basis A013610","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:            10,             12;\n  2:           150,            360,             216;\n  3:          3000,          10800,           12960,             5184;\n  4:         75000,         360000,          648000,           518400,           155520;\n  5:       2250000,       13500000,        32400000,         38880000,         23328000,          5598720;\n  6:      78750000,      567000000,      1701000000,       2721600000,       2449440000,       1175731200,        235146240;\n  7:    3150000000,    26460000000,     95256000000,     190512000000,     228614400000,     164602368000,      65840947200,     11287019520;\n  8:  141750000000,  1360800000000,   5715360000000,   13716864000000,   20575296000000,   19752284160000,   11851370496000,   4063327027200,    609499054080;\n  9: 7087500000000, 76545000000000, 367416000000000, 1028764800000000, 1851776640000000, 2222131968000000, 1777705574400000, 914248581120000, 274274574336000, 36569943244800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5*n+5+(6*n+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[(5*n + 5 + (6*n + 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] = (5*n + 5 + (6*n + 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-75a6475b3e851dc5","steps":["basis A013610","basis A013609"],"edges":["e-1eb90284df2d6b2d","e-7b1159fca4765fad"],"text":"A156992 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A156992 \u2192 basis A013610 \u2192 basis A013609","chain":null}