{"id":"n-fafe18af774a8f28","revision":0,"first_index":1,"fingerprint":"f6-2b7ed1e86e9f26f3","name":"basis_map_A013610 of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","12"],["56","192","144"],["576","2880","4320","1728"],["6016","39936","92160","82944","20736"],["62976","522240","1635840","2350080","1451520","248832"],["659456","6561792","25989120","51535872","52254720","23887872","2985984"],["6905856","80166912","383975424","972324864","1379524608","1059028992","376233984","35831808"],["72318976","959447040","5390991360","16618881024","30364139520","32997113856","20065812480","5733089280","429981696"],["757334016","11303387136","72884551680","264847491072","592514777088","835311108096","726382411776","361184624640","85136375808","5159780352"],["7930904576","131522887680","957119201280","4005801492480","10608975544320","18408057864192","20902843514880","14997761556480","6243334225920","1238347284480","61917364224"],["83053510656","1515058102272","12281645629440","58178314174464","178045971333120","367127672389632","515493398052864","485844917944320","294231314792448","104433954324480","17708366168064","743008370688"],["869747654656","17308248440832","154658987311104","817983702171648","2840705479213056","6788809387671552","11367673384402944","13304583073824768","10641365884993536","5535907700539392","1700003152134144","249650812551168","8916100448256"],["9108115685376","196358858342400","1917394196889600","11199560747581440","43517663929958400","118298510905835520","230212735126732800","322351888387276800","321426521912770560","221944855973068800","100583519834603520","27045504693043200","3477279174819840","106993205379072"],["95381425750016","2214473326657536","23459963653324800","149984799904235520","644752807852769280","1964887735869112320","4359047018887249920","7111569777780326400","8509116787646791680","7351569186784542720","4442674904243896320","1774185107863633920","421909873211473920","47932956009824256","1283918464548864"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","392","9504","241792","6271488","163874816","4293992448","112631775232","2955481841664","77563611840512","2035686674792448","53428435661357056","1402287895061987328","36804684692924137472"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["6","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","-96","-144"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","24"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203254,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-d63019265e87102e"]},{"id":203255,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-d63019265e87102e"]},{"id":203256,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-d63019265e87102e"]},{"id":203257,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-d63019265e87102e"]}],"region_exact":{},"incoming_edges":[{"id":"e-d63019265e87102e","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-fafe18af774a8f28","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":"64f8164ee6b3af95e2f2e80121bcc5e523418a358c72b276c96d74b9ff7f90e2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7ff58d52d83b38f4","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-20d71174acba6b4d","kind":"fingerprint_related","node_a_id":"n-c04eaa7a3fc005c1","node_b_id":"n-fafe18af774a8f28","fingerprint":"f6-2b7ed1e86e9f26f3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2b7ed1e86e9f26f3"},"other_id":"n-c04eaa7a3fc005c1","other":{"id":"n-c04eaa7a3fc005c1","name":"basis_map_A013610 of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","first_index":0,"fingerprint":"f6-2b7ed1e86e9f26f3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","6"],["7","24","18"]],"card_rows":[["1"],["3","6"],["7","24","18"],["18","90","135","54"]],"row_sums_preview":["1","9","49","297","1889","12249","80017","524169"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2496e2f555756362","steps":["basis A013610"],"edges":["e-087c2053836c9fd5"],"text":"A084534 \u2192 basis A013610"},"headline":"A084534 \u2192 basis A013610","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:         6,          12;\n   3:        56,         192,         144;\n   4:       576,        2880,        4320,         1728;\n   5:      6016,       39936,       92160,        82944,        20736;\n   6:     62976,      522240,     1635840,      2350080,      1451520,       248832;\n   7:    659456,     6561792,    25989120,     51535872,     52254720,     23887872,      2985984;\n   8:   6905856,    80166912,   383975424,    972324864,   1379524608,   1059028992,    376233984,     35831808;\n   9:  72318976,   959447040,  5390991360,  16618881024,  30364139520,  32997113856,  20065812480,   5733089280,   429981696;\n  10: 757334016, 11303387136, 72884551680, 264847491072, 592514777088, 835311108096, 726382411776, 361184624640, 85136375808, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16+96*x+144*x^2)*P(n-2,x) + (12+24*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 6 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(16 + 96*x + 144*x^2)*P[n - 2, x] + (12 + 24*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 12*x; for(n = 3, N, my(i = n); v[i] = -(16 + 96*x + 144*x^2)*v[i - 2] + (12 + 24*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013611","basis A013610"],"edges":["e-11143fa0a471b62a","e-d63019265e87102e"],"text":"A086645 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A086645 \u2192 basis A013611 \u2192 basis A013610","chain":null}