{"id":"n-87b7b64af43d2638","revision":1,"first_index":0,"fingerprint":"f6-6b852c5cd7d4c499","name":"basis_map_A013610 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["8","24","18"],["64","384","864","864","324"],["512","4608","17280","34560","38880","23328","5832"],["4096","49152","258048","774144","1451520","1741824","1306368","559872","104976"],["32768","491520","3317760","13271040","34836480","62705664","78382080","67184640","37791360","12597120","1889568"],["262144","4718592","38928384","194641920","656916480","1576599552","2759049216","3547348992","3325639680","2217093120","997691904","272097792","34012224"],["2097152","44040192","429391872","2576351232","10627448832","31882346496","71735279616","122974765056","161404379136","161404379136","121053284352","66029064192","24760899072","5714053632","612220032"],["16777216","402653184","4529848320","31708938240","154581073920","556491866112","1530352631808","3279327068160","5533864427520","7378485903360","7747410198528","6338790162432","3961743851520","1828497162240","587731230720","117546246144","11019960576"],["134217728","3623878656","46204452864","369635622912","2079200378880","8732641591296","28381085171712","72979933298688","150521112428544","250868520714240","338672502964224","369460912324608","323278298284032","223808052658176","119897171066880","47958868426752","13488431745024","2380311484416","198359290368"],["1073741824","32212254720","459024629760","4131221667840","26336538132480","126415383035904","474057686384640","1422173059153920","3466546831687680","6933093663375360","11439604544569344","15599460742594560","17549393335418880","16199440001925120","12149580001443840","7289748000866304","3417069375406080","1206024485437440","301506121359360","47606229688320","3570467226624"],["8589934592","283467841536","4464618504192","44646185041920","318104068423680","1717761969487872","7300488370323456","25030245841108992","70397566428119040","164260988332277760","320308927247941632","524141880951177216","720695086307868672","831571253432156160","801872280095293440","641497824076234752","420982947050029056","222873324908838912","92863885378682880","29325437488005120","6598223434801152","942603347828736","64268410079232"],["68719476736","2473901162496","42674795053056","469422745583616","3696704121470976","22180224728825856","105356067461922816","406373403067416576","1295315222277390336","3454173926073040896","7771891333664342016","14837247091541016576","24110526523754151936","33383805955967287296","39345199876675731456","39345199876675731456","33197512395945148416","23433538161843634176","13669563927742119936","6475056597351530496","2428146224006823936","693756064001949696","141904649454944256","18509302102818816","1156831381426176"],["549755813888","21440476741632","402008938905600","4824107266867200","41607925176729600","274612306166415360","1441714607373680640","6178776888744345600","22011892666151731200","66035677998455193600","168390978896060743680","367398499409587077120","688872186392975769600","1112793531865576243200","1549962419384195481600","1859954903261034577920","1918078493987941908480","1692422200577595801600","1269316650433196851200","801673673957808537600","420878678827849482240","180376576640506920960","61492014763809177600","16041395155776307200","3007761591708057600","360931391004966912","20822964865671168"],["4398046511104","184717953466368","3740538557693952","48627001250021376","455878136718950400","3282322584376442880","18873354860164546560","88974387197918576640","350336649591804395520","1167788831972681318400","3328198171122141757440","8169213692754347950080","17359579097102989393920","32048453717728595804160","51506443474920957542400","72109020864889340559360","87882869179083883806720","93052449719029994618880","85298078909110828400640","67340588612455917158400","45454897313407744081920","25974227036232996618240","12396790176383930204160","4850917895106755297280","1515911842220861030400","363818842133006647296","62968645753789612032","6996516194865512448","374813367582081024"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","50","2500","125000","6250000","312500000","15625000000","781250000000","39062500000000","1953125000000000","97656250000000000","4882812500000000000","244140625000000000000","12207031250000000000000","610351562500000000000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","24","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167419,"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-a142fb8711dcbce4"]},{"id":167420,"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-a142fb8711dcbce4"]},{"id":167421,"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-a142fb8711dcbce4"]}],"region_exact":{},"incoming_edges":[{"id":"e-a142fb8711dcbce4","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-87b7b64af43d2638","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":"35adc64c1f61ec827bd722a2c416a0278b21d1023de0673daba34ec0dd09f090","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-336459bdce627a87","role":"sequence","name":"Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","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-dd12ac9defee45ea","kind":"fingerprint_related","node_a_id":"n-1e083728b424e77f","node_b_id":"n-87b7b64af43d2638","fingerprint":"f6-6b852c5cd7d4c499","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6b852c5cd7d4c499"},"other_id":"n-1e083728b424e77f","other":{"id":"n-1e083728b424e77f","name":"basis_map_A013610 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-6b852c5cd7d4c499","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","12","9"],["16","96","216","216","81"]],"card_rows":[["1"],["4","12","9"],["16","96","216","216","81"],["64","576","2160","4320","4860","2916","729"]],"row_sums_preview":["1","25","625","15625","390625","9765625","244140625","6103515625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013610"],"edges":["e-cb7f9b19b94bd36e"],"text":"A034870 \u2192 basis A013610"},"headline":"A034870 \u2192 basis A013610","chain":null}},{"id":"l-a65af5618dd6a3fc","kind":"oeis_match","node_a_id":"n-87b7b64af43d2638","node_b_id":null,"fingerprint":null,"oeis_id":"A165800","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A165800","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A165800","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-87b7b64af43d2638","oeis_id":"A165800","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":17,"oeis_terms_available":17,"term_count":200,"raw_size":34800}],"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:         8,         24,          18;\n  2:        64,        384,         864,          864,           324;\n  3:       512,       4608,       17280,        34560,         38880,         23328,           5832;\n  4:      4096,      49152,      258048,       774144,       1451520,       1741824,        1306368,         559872,          104976;\n  5:     32768,     491520,     3317760,     13271040,      34836480,      62705664,       78382080,       67184640,        37791360,        12597120,         1889568;\n  6:    262144,    4718592,    38928384,    194641920,     656916480,    1576599552,     2759049216,     3547348992,      3325639680,      2217093120,       997691904,       272097792,        34012224;\n  7:   2097152,   44040192,   429391872,   2576351232,   10627448832,   31882346496,    71735279616,   122974765056,    161404379136,    161404379136,    121053284352,     66029064192,     24760899072,      5714053632,       612220032;\n  8:  16777216,  402653184,  4529848320,  31708938240,  154581073920,  556491866112,  1530352631808,  3279327068160,   5533864427520,   7378485903360,   7747410198528,   6338790162432,   3961743851520,   1828497162240,    587731230720,   117546246144,    11019960576;\n  9: 134217728, 3623878656, 46204452864, 369635622912, 2079200378880, 8732641591296, 28381085171712, 72979933298688, 150521112428544, 250868520714240, 338672502964224, 369460912324608, 323278298284032, 223808052658176, 119897171066880, 47958868426752, 13488431745024, 2380311484416, 198359290368;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+24*x+18*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(8 + 24*x + 18*x^2)*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] = (8 + 24*x + 18*x^2)*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-336459bdce627a87","steps":["basis A013610"],"edges":["e-a142fb8711dcbce4"],"text":"A139548 \u2192 basis A013610"},"headline":"A139548 \u2192 basis A013610","chain":null}
