{"id":"n-1bebb7994606abe7","revision":0,"first_index":0,"fingerprint":"f6-abfa4e49247086e7","name":"basis_map_A013611 of derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["2","8"],["4","48","192","256"],["6","120","960","3840","7680","6144"],["8","224","2688","17920","71680","172032","229376","131072"],["10","360","5760","53760","322560","1290240","3440640","5898240","5898240","2621440"],["12","528","10560","126720","1013760","5677056","22708224","64880640","129761280","173015040","138412032","50331648"],["14","728","17472","256256","2562560","18450432","98402304","393609216","1180827648","2624061440","4198498304","4580179968","3053453312","939524096"],["16","960","26880","465920","5591040","49201152","328007680","1686896640","6747586560","20992491520","50381979648","91603599360","122138132480","112742891520","64424509440","17179869184"],["18","1224","39168","783360","10967040","114057216","912457728","5735448576","28677242880","114708971520","367068708864","934356713472","1868713426944","2874943733760","3285649981440","2628519985152","1314259992576","309237645312"],["20","1520","54720","1240320","19845120","238141440","2222653440","16511139840","99066839040","484326768640","1937307074560","6340277698560","16907407196160","36415953960960","62427349647360","83236466196480","83236466196480","58755152609280","26113401159680","5497558138880"],["22","1848","73920","1872640","33707520","458422272","4889837568","41912893440","293390254080","1695143690240","8136689713152","32546758852608","108489196175360","300431620177920","686700846120960","1281841579425792","1922762369138688","2262073375457280","2010731889295360","1269935930081280","507974372032512","96757023244288"],["24","2208","97152","2720256","54405120","826957824","9923493888","96399654912","771197239296","5141314928640","28791363600384","136104627929088","544418511716352","1842647270424576","5264706486927360","12635295568625664","25270591137251328","41622150108413952","55496200144551936","58417052783738880","46733642226991104","26704938415423488","9710886696517632","1688849860263936"],["26","2600","124800","3827200","84198400","1414533120","18860441600","204770508800","1842934579200","13924394598400","89116125429760","486087956889600","2268410465484800","9073641861939200","31109629240934400","91254912440074240","228137281100185600","483114242329804800","858869764141875200","1265702810314342400","1518843372377210880","1446517497502105600","1052012725456076800","548876204585779200","182958734861926400","29273397577908224"],["28","3024","157248","5241600","125798400","2314690560","33948794880","407385538560","4073855385600","34401445478400","247690407444480","1531177064202240","8166277675745280","37690512349593600","150762049398374400","522641771247697920","1567925313743093760","4058159635570360320","9018132523489689600","17086987939243622400","27339180702789795840","36452240937053061120","39766081022239703040","34579200888904089600","23052800592602726400","11065344284449308672","3404721318292094976","504403158265495552"]],"degrees":[-1,1,3,5,7,9,11,13,15,17,19,21,23,25,27],"row_sums":["0","10","500","18750","625000","19531250","585937500","17089843750","488281250000","13732910156250","381469726562500","10490417480468750","286102294921875000","7748603820800781250","208616256713867187500"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2","8"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","8","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204156,"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-bad944fbc6d5efd9"]},{"id":204157,"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-bad944fbc6d5efd9"]},{"id":204158,"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-bad944fbc6d5efd9"]}],"region_exact":{},"incoming_edges":[{"id":"e-bad944fbc6d5efd9","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-1bebb7994606abe7","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":"27ac471175b3aa0fe056576ecae935e441f9e2fca4d75385533922f7340693e2","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-e75bc09e82ca4254","role":"sequence","name":"derivative of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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-401b4b882888bc3e","kind":"fingerprint_related","node_a_id":"n-a9faf912d2e8db2a","node_b_id":"n-1bebb7994606abe7","fingerprint":"f6-abfa4e49247086e7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-abfa4e49247086e7"},"other_id":"n-a9faf912d2e8db2a","other":{"id":"n-a9faf912d2e8db2a","name":"derivative of basis_map_A013611 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-abfa4e49247086e7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["8","32"],["16","192","768","1024"]],"card_rows":[["8","32"],["16","192","768","1024"],["24","480","3840","15360","30720","24576"],["32","896","10752","71680","286720","688128","917504","524288"]],"row_sums_preview":["0","40","2000","75000","2500000","78125000","2343750000","68359375000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A013611","D"],"edges":["e-a80b60183a85d18e","e-e2c579bba83467a1"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013611 \u2192 D"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013611 \u2192 D","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:  0;\n  1:  2,    8;\n  2:  4,   48,   192,    256;\n  3:  6,  120,   960,   3840,     7680,      6144;\n  4:  8,  224,  2688,  17920,    71680,    172032,    229376,     131072;\n  5: 10,  360,  5760,  53760,   322560,   1290240,   3440640,    5898240,     5898240,      2621440;\n  6: 12,  528, 10560, 126720,  1013760,   5677056,  22708224,   64880640,   129761280,    173015040,    138412032,     50331648;\n  7: 14,  728, 17472, 256256,  2562560,  18450432,  98402304,  393609216,  1180827648,   2624061440,   4198498304,   4580179968,    3053453312,     939524096;\n  8: 16,  960, 26880, 465920,  5591040,  49201152, 328007680, 1686896640,  6747586560,  20992491520,  50381979648,  91603599360,  122138132480,  112742891520,   64424509440,   17179869184;\n  9: 18, 1224, 39168, 783360, 10967040, 114057216, 912457728, 5735448576, 28677242880, 114708971520, 367068708864, 934356713472, 1868713426944, 2874943733760, 3285649981440, 2628519985152, 1314259992576, 309237645312;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n+8*n*x+16*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 8*n*x + 16*n*x^2)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 8*n*x + 16*n*x^2)*v[i - 1])/(n - 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-cdb0db4ad473ee28","steps":["D","basis A013611"],"edges":["e-521a2a476289c028","e-bad944fbc6d5efd9"],"text":"basis-map coefficients of A297191, shifted \u2192 D \u2192 basis A013611"},"headline":"basis-map coefficients of A297191, shifted \u2192 D \u2192 basis A013611","chain":null}