{"id":"n-1600c9e1b40ceecc","revision":1,"first_index":0,"fingerprint":"f6-89a932e93fe55d02","name":"basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","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":[["121","352","256"],["1331","5808","8448","4096"],["14641","85184","185856","180224","65536"],["161051","1171280","3407360","4956160","3604480","1048576"],["1771561","15460896","56221440","109035520","118947840","69206016","16777216"],["19487171","198414832","865810176","2098933760","3052994560","2664431616","1291845632","268435456"],["214358881","2494357888","12698549248","36941234176","67165880320","78156660736","56841207808","23622320128","4294967296"],["2357947691","30867678864","179593767936","609530363904","1329884430336","1934377353216","1875759857664","1169304846336","425201762304","68719476736"],["25937424601","377271630560","2469414309120","9578334289920","24381214556160","42556301770752","51583396085760","42874511032320","23386096926720","7559142440960","1099511627776"],["285311670611","4564986729776","33199903489280","144872306135040","421446708756480","858218752376832","1248318185275392","1296953958727680","943239242711040","457328117678080","133040906960896","17592186044416"],["3138428376721","54779840757312","438238726058496","2124793823313920","6953870694481920","16183553616248832","27463000076058624","34239584510410752","31126895009464320","20122437177835520","8780699859419136","2322168557862912","281474976710656"],["34522712143931","652793102357968","5697103438760448","30384551673389056","110489278812323840","289281020890447872","561029858696626176","816043430831456256","890229197270679552","719377129107619840","418546693298978816","166035051887198208","40250921669623808","4503599627370496"],["379749833583241","7733087520240544","73112827464092416","425383723427446784","1701534893709787136","4949919690792108032","10799824779910053888","17952955478292037632","22849216063280775168","22156815576514691072","16114047692010684416","8523132663542841344","3099320968561033216","693554342615056384","72057594037927936"],["4177248169415651","91139960059977840","927970502428865280","5849026197127393280","25523023405646807040","81673674898069782528","197996787631684321280","370279706739773276160","538588664348761128960","609312428354154004480","531763573836352585728","351579222371142205440","170462653270856826880","57218233265742151680","11889503016258109440","1152921504606846976"],["45949729863572161","1069375531370406656","11665914887677163520","79186816207263170560","374337676616153169920","1306778798369116520448","3484743462317644054528","7241025376244455178240","11848950615672744837120","15319855341475872112640","15598398165866342514688","12375588627464205631488","7500356743917700382720","3356803018256872898560","1046276265430713630720","202914184810805067776","18446744073709551616"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987","205891132094649","5559060566555523","150094635296999121","4052555153018976267","109418989131512359209","2954312706550833698643","79766443076872509863361"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["121","352","256"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["11","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166836,"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-c8a2ee9cf24a53ed"]},{"id":166837,"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-c8a2ee9cf24a53ed"]},{"id":166838,"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-c8a2ee9cf24a53ed"]}],"region_exact":{},"incoming_edges":[{"id":"e-c8a2ee9cf24a53ed","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-1600c9e1b40ceecc","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":"c89d5c5ac5fed2f9770ccacc5fd72e30f5c976f50f5fb964ef93c146d46246c9","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-26633b980c3508e8","role":"sequence","name":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","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-fc80c78bdb78fde1","kind":"oeis_match","node_a_id":"n-1600c9e1b40ceecc","node_b_id":null,"fingerprint":null,"oeis_id":"A009971","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":"A009971","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009971","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:          121,           352,            256;\n  1:         1331,          5808,           8448,            4096;\n  2:        14641,         85184,         185856,          180224,           65536;\n  3:       161051,       1171280,        3407360,         4956160,         3604480,         1048576;\n  4:      1771561,      15460896,       56221440,       109035520,       118947840,        69206016,         16777216;\n  5:     19487171,     198414832,      865810176,      2098933760,      3052994560,      2664431616,       1291845632,        268435456;\n  6:    214358881,    2494357888,    12698549248,     36941234176,     67165880320,     78156660736,      56841207808,      23622320128,      4294967296;\n  7:   2357947691,   30867678864,   179593767936,    609530363904,   1329884430336,   1934377353216,    1875759857664,    1169304846336,    425201762304,     68719476736;\n  8:  25937424601,  377271630560,  2469414309120,   9578334289920,  24381214556160,  42556301770752,   51583396085760,   42874511032320,  23386096926720,   7559142440960,   1099511627776;\n  9: 285311670611, 4564986729776, 33199903489280, 144872306135040, 421446708756480, 858218752376832, 1248318185275392, 1296953958727680, 943239242711040, 457328117678080, 133040906960896, 17592186044416;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 352*x + 256*x^2.","mathematica":"P[0, x_] := 121 + 352*x + 256*x^2;\nP[n_, x_] := P[n, x] = Expand[(11 + 16*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] = 121 + 352*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 16*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-26633b980c3508e8","steps":["basis A013609"],"edges":["e-c8a2ee9cf24a53ed"],"text":"A038226 \u2192 basis A013609"},"headline":"A038226 \u2192 basis A013609","chain":null}