{"id":"n-34a4e524e6a5040f","revision":0,"first_index":0,"fingerprint":"f6-52cac31f6d7fd0d8","name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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":true,"rows":[["1"],["4","2"],["40","44","12"],["760","1316","756","144"],["23560","55996","49756","19584","2880"],["1083760","3282616","3968656","2393544","720000","86400"],["69360640","255605344","391863856","319870368","146608848","35769600","3628800"],["5895654400","25610650080","47622327024","49133357216","30374492688","11250511488","2311545600","203212800"],["642626329600","3216047975520","7034800451376","8784343482272","6848421422544","3413269225728","1061995297536","188581478400","14631321600"],["87397180825600","495218894334720","1246177179183936","1827802754212832","1721976226870464","1080562542727968","451625590780416","121226657840640","18962192793600","1316818944000"],["14507932017049600","91820026350379520","261339490121352576","440494746909563072","486906356623908544","368790767048593728","193831727769625536","69802440187392000","16482656366208000","2304433152000000","144850083840000"],["2887078471392870400","20187232269976071680","64126802012399259264","122155267331021591360","155039671560220125760","137661001717026079680","87252895076569853760","39476473662881578752","12493970721611136000","2634292837587456000","333010342748160000","19120211066880000"],["678463440777324544000","5194383824981664627200","18219006707030093109120","38710268936724358414784","55490544520291097805760","56536524166895468343360","41979546610849984063680","22888422942722068193280","9094413010988143245312","2568118249404389376000","489207113209460736000","56442863069429760000","2982752926433280000"],["185898982772986925056000","1546741514266449174860800","5937385693872908474049280","13922472909341951151510656","22249678145043594030268928","25590286724422338126728960","21792043169747870871939840","13911705369480543784548480","6657562140586167660392448","2358847568336644759670784","601440270410991108096000","104501039085145608192000","11089875380478935040000","542861032610856960000"],["58744078556263868317696000","527809104890525193517772800","2201029597259793404520340480","5646352435065367343427716096","9954617604795585455382219008","12758963015376613594402826240","12260245853769018202146071040","8972427962402904719024686080","5025247764016143175439194368","2143483881117474952738381824","685413114800568589689200640","159324785137214144888832000","25449618828111921192960000","2500417916205607157760000","114000816848279961600000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","96","2976","151776","11534976","1222707456","172401751296","31204716984576","7052266038514176","1946425426629912576","644266816214501062656","251908325139869915498496","114870196263780681467314176","60421723234748638451807256576"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"]]},"terms":[{"coeff":{"coeffs":[["2","0"],["3","2"],["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-8f4e693782942de7","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["3/2","1"],["3/2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192257,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-09adfa810b7ca037"]},{"id":192258,"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-09adfa810b7ca037","e-4b5346f304dc9a08"]},{"id":192259,"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-09adfa810b7ca037","e-4b5346f304dc9a08"]},{"id":192260,"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-09adfa810b7ca037","e-4b5346f304dc9a08"]},{"id":192261,"position":4,"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-4b5346f304dc9a08"]}],"region_exact":{},"incoming_edges":[{"id":"e-4b5346f304dc9a08","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-34a4e524e6a5040f","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":"2bc96f9086ceb0a37223365d424acf919463713135241dcba0320b82cd0be1d1","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-623036dc25b33b8c","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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]"}]},{"id":"e-8f4e693782942de7","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-34a4e524e6a5040f","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"e23b2a45cf21248227d276e1d7696d1bf804bc602abe525de48e97700ca27d1d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-09adfa810b7ca037","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-34a4e524e6a5040f","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"8cb0b278e3dd85edac0d4738ed7c9455db95ec0f66974a28d36c08024e204819","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-623036dc25b33b8c","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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:              4,               2;\n  2:             40,              44,               12;\n  3:            760,            1316,              756,              144;\n  4:          23560,           55996,            49756,            19584,             2880;\n  5:        1083760,         3282616,          3968656,          2393544,           720000,            86400;\n  6:       69360640,       255605344,        391863856,        319870368,        146608848,         35769600,         3628800;\n  7:     5895654400,     25610650080,      47622327024,      49133357216,      30374492688,      11250511488,      2311545600,       203212800;\n  8:   642626329600,   3216047975520,    7034800451376,    8784343482272,    6848421422544,    3413269225728,   1061995297536,    188581478400,    14631321600;\n  9: 87397180825600, 495218894334720, 1246177179183936, 1827802754212832, 1721976226870464, 1080562542727968, 451625590780416, 121226657840640, 18962192793600, 1316818944000;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (3*n^2+3*n+2+(2*n^2+2*n)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 + 3*n + 2 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(2)]];\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] = ((3*n^2 + 3*n + 2 + (2*n^2 + 2*n)*x)*v[i - 1])/(2)); 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-ecd77b030b8fab94","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-42bb566932c0b2ac","e-09adfa810b7ca037"],"text":"A128813 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A128813 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}