{"id":"n-6a1d0871f44aed3c","revision":1,"first_index":0,"fingerprint":"f6-ce583c2d05188b3b","name":"basis_map_A007318 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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"],["2","1"],["10","13","4"],["100","220","157","36"],["1700","5340","6189","3124","576"],["44200","181340","294414","235949","93076","14400"],["1635400","8300780","17421558","19329017","11937976","3883536","518400"],["81770000","495173600","1277816120","1820107192","1544020633","779137624","216213264","25401600"],["5315050000","37419564000","114749158200","200087199160","216848201433","149461266072","63918670096","15488752896","1625702400"],["435834100000","3498923298000","12440415656400","25701832145320","33988615649466","29820528133977","17347693499704","6447490015248","1387896581376","131681894400"],["44019244100000","396974663098000","1606374311096400","3839926612317320","6003033395128066","6410734906478277","4734169856867804","2385965841510448","784926556243776","152089529472000","13168189440000"],["5370347780200000","53757237434056000","244011600188618800","662842338345377440","1197001194296019772","1508476699400845780","1353267646221743605","863922385345278940","384462906684504880","113531035901080896","20009352177792000","1593350922240000"],["778700428129000000","8568129508286920000","43122724217853790000","131249809487240836000","269014469894657218300","391097293391749485268","413444453415874615045","320139286930996525420","180151944958973374960","71824658768225432640","19249825235535489024","3112382597326848000","229442532802560000"],["132379072781930000000","1588182388762577400000","8778877003935633780000","29600208005648232630000","67913677685435428395000","111949985288794482388260","136380999663904347567942","124295791405552219264005","84729370134363886539180","42655870688664823917040","15410837621871131250240","3782325506351061805056","564997889524672512000","38775788043632640000"],["26078677338040210000000","338818228851486027800000","2040722517972785025060000","7551900869884086048990000","19180635273137832989295000","35365227928237856995907220","48809254050392875018983534","51216946841019039318325617","41053661031957920623963440","25010163072002292073336160","11396485666486918344037120","3765642298637900900643072","852640383481168598654976","118378416591431442432000","7600054456551997440000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","27","513","16929","863379","63026667","6239640033","804913564257","131200910973891","26371383105752091","6408246094697758113","1851983121367652094657","627822278143634060088723","246734155310448185614868139"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","0"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161996,"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-0b3719baa0dde23e"]},{"id":161997,"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-0b3719baa0dde23e"]},{"id":161998,"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-0b3719baa0dde23e"]},{"id":161999,"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-0b3719baa0dde23e"]}],"region_exact":{},"incoming_edges":[{"id":"e-0b3719baa0dde23e","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6a1d0871f44aed3c","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":"3981d079ea6ba1002d975cf5e0952ba93b12cdfe95c5a7575f35034b10a47379","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-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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-cfcb32920569a1d3","kind":"oeis_match","node_a_id":"n-6a1d0871f44aed3c","node_b_id":null,"fingerprint":null,"oeis_id":"A277352","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A277352","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A277352","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-6a1d0871f44aed3c","oeis_id":"A277352","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":16,"oeis_terms_available":16,"term_count":200,"raw_size":73183}],"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:            2,             1;\n  2:           10,            13,              4;\n  3:          100,           220,            157,             36;\n  4:         1700,          5340,           6189,           3124,            576;\n  5:        44200,        181340,         294414,         235949,          93076,          14400;\n  6:      1635400,       8300780,       17421558,       19329017,       11937976,        3883536,         518400;\n  7:     81770000,     495173600,     1277816120,     1820107192,     1544020633,      779137624,      216213264,      25401600;\n  8:   5315050000,   37419564000,   114749158200,   200087199160,   216848201433,   149461266072,    63918670096,   15488752896,    1625702400;\n  9: 435834100000, 3498923298000, 12440415656400, 25701832145320, 33988615649466, 29820528133977, 17347693499704, 6447490015248, 1387896581376, 131681894400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2+1+n^2*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[(n^2 + 1 + n^2*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; for(n = 1, N, my(i = n + 1); v[i] = (n^2 + 1 + n^2*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-e734a631f8279507","steps":["basis A007318"],"edges":["e-0b3719baa0dde23e"],"text":"A008955 \u2192 basis A007318"},"headline":"A008955 \u2192 basis A007318","chain":null}
