{"id":"n-40176f9b9853cb34","revision":0,"first_index":0,"fingerprint":"f6-edc8f1d0e47ba8f5","name":"basis_map_A013610 of basis_map_A013609 of Triangular array formed by the Mersenne numbers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","6"],["34","72","36"],["192","624","648","216"],["1084","4752","7560","5184","1296"],["6120","33768","72576","76032","38880","7776"],["34552","229824","622944","881280","686880","279936","46656"],["195072","1518720","4971456","8873280","9331200","5785344","1959552","279936"],["1101328","9823104","37695168","81305856","107853120","90139392","46376064","13436928","1679616"],["6217824","62509152","275166720","696259584","1116291456","1176384384","815173632","358318080","90699264","10077696"],["35104288","392715648","1950664896","5665946112","10659600000","13575776256","11856595968","7014076416","2690744832","604661760","60466176"],["198190080","2441901312","13509949824","44307146880","95720693760","143059488768","150963886080","112507398144","58047528960","19752284160","3990767616","362797056"],["1118931904","15055117056","91809777024","335570688000","818847843840","1405529542656","1740427057152","1566799552512","1017948072960","465589555200","142337378304","26121388032","2176782336"],["6317211264","92160491136","614169464832","2475668496384","6735069803520","13060145341440","18573811826688","19618344861696","15392390694912","8861721200640","3639580065792","1010027003904","169789022208","13060694016"],["35665403776","560755980288","4054360181760","17867886391296","53626734111744","115960231784448","186322888894464","226019341025280","208028517193728","144593492262912","74723132841984","27845399642112","7074542592000","1097098297344","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","142","1680","19876","235152","2782072","32914560","389410576","4607097792","54506352352","644862032640","7629331686976","90262256178432","1067888410767232"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202640,"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-d1e2fb5d0b24beb2"]},{"id":202641,"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-d1e2fb5d0b24beb2"]},{"id":202642,"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-d1e2fb5d0b24beb2"]},{"id":202643,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-d1e2fb5d0b24beb2"]}],"region_exact":{},"incoming_edges":[{"id":"e-d1e2fb5d0b24beb2","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-40176f9b9853cb34","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":"99e53bd46e53d49784047949d2451563359c9b14bc4ee987dfa4b07b3075275a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-107a00b591d3c4c8","role":"sequence","name":"basis_map_A013609 of Triangular array formed by the Mersenne numbers.","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":[],"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:       1;\n  1:       6,        6;\n  2:      34,       72,        36;\n  3:     192,      624,       648,       216;\n  4:    1084,     4752,      7560,      5184,       1296;\n  5:    6120,    33768,     72576,     76032,      38880,       7776;\n  6:   34552,   229824,    622944,    881280,     686880,     279936,     46656;\n  7:  195072,  1518720,   4971456,   8873280,    9331200,    5785344,   1959552,    279936;\n  8: 1101328,  9823104,  37695168,  81305856,  107853120,   90139392,  46376064,  13436928,  1679616;\n  9: 6217824, 62509152, 275166720, 696259584, 1116291456, 1176384384, 815173632, 358318080, 90699264, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -2*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-2*P[n - 2, x] + (6 + 6*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] = 1; v[2] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (6 + 6*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-9b5c7e97e5191da6","steps":["basis A013609","basis A013610"],"edges":["e-3133312db52cd60e","e-d1e2fb5d0b24beb2"],"text":"A110441 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A110441 \u2192 basis A013609 \u2192 basis A013610","chain":null}
