{"id":"n-43a1e62341e55229","revision":1,"first_index":0,"fingerprint":"f6-57ca34e86b523c55","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+11x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["144","528","484"],["1728","9504","17424","10648"],["20736","152064","418176","511104","234256"],["248832","2280960","8363520","15333120","14055360","5153632"],["2985984","32845824","150543360","367994880","505992960","371061504","113379904"],["35831808","459841536","2529128448","7727892480","14167802880","15584583168","9523911936","2494357888"],["429981696","6306398208","40466055168","148375535616","340027269120","498706661376","457147772928","239458357248","54875873536"],["5159780352","85136375808","624333422592","2670759641088","7344589012992","13465079857152","16457319825408","12930751291392","5926594341888","1207269217792"],["61917364224","1135151677440","9365001338880","45784450990080","146891780259840","323161916571648","493719594762240","517230051655680","355595660513280","144872306135040","26559922791424"],["743008370688","14984002142208","137353352970240","755443441336320","2769959284899840","7109562164576256","13034197301723136","17068591704637440","15646209062584320","9561572204912640","3505909808467968","584318301411328"],["8916100448256","196154209861632","1977888282771456","12087095061381120","49859267128197120","146253850242711552","312820735241355264","491575441093558272","563263526253035520","458955465835806720","252425506209693696","84141835403231232","12855002631049216"],["106993205379072","2550004728201216","28050052010213376","188558682957545472","864227296888750080","2851950079732875264","6971433528235917312","12780961468432515072","17573822019094708224","17899263167596462080","13126126322904072192","6563063161452036096","2005380410443677696","282810057883082752"],["1283918464548864","32953907256754176","392700728142987264","2879805339715239936","14519018587731001344","53236401488347004928","146400104092954263552","306743075242380361728","492067016534651830272","601415242431241125888","551297305561971032064","367531537041314021376","168451954477268926464","47512089724357902336","6221821273427820544"],["15407021574586368","423693093301125120","5437394697364439040","43197080095728599040","237583940526507294720","958255226790246088704","2928002081859085271040","6901719192953558138880","12653151853748189921280","18042457272937233776640","19846703000230957154304","16538919166859130961920","10107117268636135587840","4276088075192211210240","1119927829217007697920","136880068015412051968"],["184884258895036416","5423271594254401536","74569984420998021120","637987644490760847360","3801343048424116715520","16725909413066113548288","56217639971694437203968","147236676116342573629440","303675644489956558110720","494878828057706983587840","635094496007390628937728","635094496007390628937728","485141628894534508216320","273669636812301517455360","107513071604832739000320","26280973058959113977856","3011361496339065143296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1156","39304","1336336","45435424","1544804416","52523350144","1785793904896","60716992766464","2064377754059776","70188843638032384","2386420683693101056","81138303245565435904","2758702310349224820736","93795878551873643905024","3189059870763703892770816"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["144","528","484"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["12","22"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166857,"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-4aabe19e5745ac70"]},{"id":166858,"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-4aabe19e5745ac70"]},{"id":166859,"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-4aabe19e5745ac70"]}],"region_exact":{},"incoming_edges":[{"id":"e-4aabe19e5745ac70","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-43a1e62341e55229","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":"878f81159eda42b4267a1e510b7b44c55eb14be58e7fdd922b00428b0d49f6b9","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-b1e393c17b0898c2","role":"sequence","name":"Triangle of coefficients in expansion of (1+11x)^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-6fc4554ee418cd7f","kind":"fingerprint_related","node_a_id":"n-43a1e62341e55229","node_b_id":"n-196d480a741a9e86","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-196d480a741a9e86","other":{"id":"n-196d480a741a9e86","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","33"],["324","1188","1089"]],"card_rows":[["1"],["18","33"],["324","1188","1089"],["5832","32076","58806","35937"]],"row_sums_preview":["1","51","2601","132651","6765201","345025251","17596287801","897410677851"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b7cf68162d40a956","steps":["basis A013610"],"edges":["e-77c9a747c1207e17"],"text":"A038277 \u2192 basis A013610"},"headline":"A038277 \u2192 basis A013610","chain":null}},{"id":"l-a9b14ec30a5e7cd8","kind":"fingerprint_related","node_a_id":"n-b536b63a5727c887","node_b_id":"n-43a1e62341e55229","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-b536b63a5727c887","other":{"id":"n-b536b63a5727c887","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","528","144"],["10648","17424","9504","1728"],["234256","511104","418176","152064","20736"]],"card_rows":[["484","528","144"],["10648","17424","9504","1728"],["234256","511104","418176","152064","20736"],["5153632","14055360","15333120","8363520","2280960","248832"]],"row_sums_preview":["1156","39304","1336336","45435424","1544804416","52523350144","1785793904896","60716992766464"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["basis A007318"],"edges":["e-f9a8e29d679c9a8f"],"text":"A038314 \u2192 basis A007318"},"headline":"A038314 \u2192 basis A007318","chain":null}},{"id":"l-e8bcbdc3eb6cf90b","kind":"fingerprint_related","node_a_id":"n-1f2dec04776afb2b","node_b_id":"n-43a1e62341e55229","fingerprint":"f6-57ca34e86b523c55","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-57ca34e86b523c55"},"other_id":"n-1f2dec04776afb2b","other":{"id":"n-1f2dec04776afb2b","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","first_index":0,"fingerprint":"f6-57ca34e86b523c55","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","132","36"],["1331","2178","1188","216"],["14641","31944","26136","9504","1296"]],"card_rows":[["121","132","36"],["1331","2178","1188","216"],["14641","31944","26136","9504","1296"],["161051","439230","479160","261360","71280","7776"]],"row_sums_preview":["289","4913","83521","1419857","24137569","410338673","6975757441","118587876497"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1f2dec04776afb2b","steps":[],"edges":[],"text":"A038320"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","chain":null}},{"id":"l-76e126480a48d80e","kind":"oeis_match","node_a_id":"n-43a1e62341e55229","node_b_id":null,"fingerprint":null,"oeis_id":"A009978","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009978","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009978","relation":"match_row_sums"}],"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:          144,            528,             484;\n  1:         1728,           9504,           17424,           10648;\n  2:        20736,         152064,          418176,          511104,           234256;\n  3:       248832,        2280960,         8363520,        15333120,         14055360,          5153632;\n  4:      2985984,       32845824,       150543360,       367994880,        505992960,        371061504,         113379904;\n  5:     35831808,      459841536,      2529128448,      7727892480,      14167802880,      15584583168,        9523911936,        2494357888;\n  6:    429981696,     6306398208,     40466055168,    148375535616,     340027269120,     498706661376,      457147772928,      239458357248,       54875873536;\n  7:   5159780352,    85136375808,    624333422592,   2670759641088,    7344589012992,   13465079857152,    16457319825408,    12930751291392,     5926594341888,    1207269217792;\n  8:  61917364224,  1135151677440,   9365001338880,  45784450990080,  146891780259840,  323161916571648,   493719594762240,   517230051655680,   355595660513280,  144872306135040,   26559922791424;\n  9: 743008370688, 14984002142208, 137353352970240, 755443441336320, 2769959284899840, 7109562164576256, 13034197301723136, 17068591704637440, 15646209062584320, 9561572204912640, 3505909808467968, 584318301411328;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+22*x)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 528*x + 484*x^2.","mathematica":"P[0, x_] := 144 + 528*x + 484*x^2;\nP[n_, x_] := P[n, x] = Expand[(12 + 22*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] = 144 + 528*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = (12 + 22*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-b1e393c17b0898c2","steps":["basis A013609"],"edges":["e-4aabe19e5745ac70"],"text":"A013618 \u2192 basis A013609"},"headline":"A013618 \u2192 basis A013609","chain":null}