{"id":"n-058123003a903c80","revision":1,"first_index":0,"fingerprint":"f6-c62c9860d860e562","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^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":[["484","1320","900"],["10648","43560","59400","27000"],["234256","1277760","2613600","2376000","810000"],["5153632","35138400","95832000","130680000","89100000","24300000"],["113379904","927653760","3162456000","5749920000","5880600000","3207600000","729000000"],["2494357888","23809779840","97403644800","221371920000","301870800000","246985200000","112266000000","21870000000"],["54875873536","598645893120","2857173580800","7792291584000","13282315200000","14489798400000","9879408000000","3849120000000","656100000000"],["1207269217792","14816485854720","80817195571200","257145622272000","525979681920000","717245020800000","652040928000000","381062880000000","129907800000000","19683000000000"],["26559922791424","362180765337600","2222472878208000","8081719557120000","19285921670400000","31558780915200000","35862251040000000","27944611200000000","14289858000000000","4330260000000000","590490000000000"],["584318301411328","8764774521169920","59759826280704000","244472016602880000","666741863462400000","1272870830246400000","1735732950336000000","1690648977600000000","1152715212000000000","523961460000000000","142898580000000000","17714700000000000"],["12855002631049216","210354588508078080","1577659413810585600","7171179153684480000","22002481494259200000","48005414169292800000","76372249814784000000","89266266017280000000","76079203992000000000","46108608480000000000","18862612560000000000","4676680800000000000","531441000000000000"],["282810057883082752","5013451026109194240","41019144759075225600","205095723795376128000","699189967484236800000","1716193556552217600000","3120351921004032000000","4255025346823680000000","4351730468342400000000","3296765506320000000000","1798235730720000000000","668765354400000000000","151992126000000000000","15943230000000000000"],["6221821273427820544","118780224310894755840","1052824715482930790400","5742680266270531584000","21535050998514493440000","58731957268675891200000","120133548958655232000000","187221115260241920000000","223388830708243200000000","203080755189312000000000","138464151265440000000000","68659909718400000000000","23406787404000000000000","4910514840000000000000","478296900000000000000"],["136880068015412051968","2799819573042519244800","26725550469951320064000","157923707322439618560000","646051529955434803200000","1938154589866304409600000","4404896795150691840000000","7722871004484979200000000","10531187733388608000000000","11169441535412160000000000","9138633983519040000000000","5664442551768000000000000","2574746614440000000000000","810234948600000000000000","157837977000000000000000","14348907000000000000000"],["3011361496339065143296","65702432647397784944640","671956697530204618752000","4276088075192211210240000","18950844878692754227200000","62020946875721741107200000","155052367189304352768000000","302050065953190297600000000","463372260269098752000000000","561663345780725760000000000","536133193699783680000000000","398776755644467200000000000","226577702070720000000000000","95067567302400000000000000","27779483952000000000000000","5050815264000000000000000","430467210000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2704","140608","7311616","380204032","19770609664","1028071702528","53459728531456","2779905883635712","144555105949057024","7516865509350965248","390877006486250192896","20325604337285010030592","1056931425538820521590784","54960434128018667122720768","2857942574656970690381479936"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["484","1320","900"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["22","30"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167600,"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-022a6c0efe3a8d25"]},{"id":167601,"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-022a6c0efe3a8d25"]},{"id":167602,"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-022a6c0efe3a8d25"]}],"region_exact":{},"incoming_edges":[{"id":"e-022a6c0efe3a8d25","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-058123003a903c80","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":"abf3d1760d633f547e6e13e4b149cf60a12bc155bb6aea13626ca1bd49fa1882","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-ec89fb3fc20684fc","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^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-4e411db0c8c498fb","kind":"fingerprint_related","node_a_id":"n-f562ab3e3b669a1b","node_b_id":"n-058123003a903c80","fingerprint":"f6-c62c9860d860e562","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c62c9860d860e562"},"other_id":"n-f562ab3e3b669a1b","other":{"id":"n-f562ab3e3b669a1b","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-c62c9860d860e562","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","15"],["121","330","225"]],"card_rows":[["1"],["11","15"],["121","330","225"],["1331","5445","7425","3375"]],"row_sums_preview":["1","26","676","17576","456976","11881376","308915776","8031810176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["basis A013610"],"edges":["e-aef71e51dba5316b"],"text":"A038259 \u2192 basis A013610"},"headline":"A038259 \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:             484,             1320,               900;\n  1:           10648,            43560,             59400,              27000;\n  2:          234256,          1277760,           2613600,            2376000,             810000;\n  3:         5153632,         35138400,          95832000,          130680000,           89100000,            24300000;\n  4:       113379904,        927653760,        3162456000,         5749920000,         5880600000,          3207600000,           729000000;\n  5:      2494357888,      23809779840,       97403644800,       221371920000,       301870800000,        246985200000,        112266000000,         21870000000;\n  6:     54875873536,     598645893120,     2857173580800,      7792291584000,     13282315200000,      14489798400000,       9879408000000,       3849120000000,        656100000000;\n  7:   1207269217792,   14816485854720,    80817195571200,    257145622272000,    525979681920000,     717245020800000,     652040928000000,     381062880000000,     129907800000000,     19683000000000;\n  8:  26559922791424,  362180765337600,  2222472878208000,   8081719557120000,  19285921670400000,   31558780915200000,   35862251040000000,   27944611200000000,   14289858000000000,   4330260000000000,    590490000000000;\n  9: 584318301411328, 8764774521169920, 59759826280704000, 244472016602880000, 666741863462400000, 1272870830246400000, 1735732950336000000, 1690648977600000000, 1152715212000000000, 523961460000000000, 142898580000000000, 17714700000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (22+30*x)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1320*x + 900*x^2.","mathematica":"P[0, x_] := 484 + 1320*x + 900*x^2;\nP[n_, x_] := P[n, x] = Expand[(22 + 30*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] = 484 + 1320*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = (22 + 30*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-ec89fb3fc20684fc","steps":["basis A013610"],"edges":["e-022a6c0efe3a8d25"],"text":"A038336 \u2192 basis A013610"},"headline":"A038336 \u2192 basis A013610","chain":null}