{"id":"n-702478000e7a23c3","revision":0,"first_index":0,"fingerprint":"f6-7d88dc66b737aeee","name":"reversal of basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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"],["4","6","2"],["8","32","34","10"],["16","144","388","360","100"],["32","560","3224","7316","6320","1700"],["64","1952","21008","98456","202856","167720","44200"],["128","6272","114240","974208","4048584","7841112","6294040","1635400"],["256","18944","542080","7660416","56807568","218111424","404643680","317972800","81770000"],["512","54528","2315520","50556032","611542176","4128714768","14986529920","26937784800","20831772000","5315050000"],["1024","151040","9102336","290984704","5368678976","58403887968","368527670816","1282771023040","2250561897600","1718835404000","435834100000"],["2048","405504","33459712","1501305344","40126813056","659044352512","6635848026400","39786836798496","134060997122240","230744422465600","174474044004000","44019244100000"],["4096","1060864","116390912","7084695552","263412878080","6213559897856","93675107059264","889147132817792","5122116083660992","16816930493844480","28499767628811200","21373871856688000","5370347780200000"],["8192","2715648","386607104","31046073344","1554106611200","50621987117312","1088316399307648","15361184789228864","139170566425901824","776340693118532800","2495454456865072000","4175214049891000000","3109952114780160000","778700428129000000"],["16384","6823936","1234874368","127815354368","8386045690880","365442098138624","10782370608558336","215736157460757888","2889742547020710528","25211677678640375680","136968826743880720000","432577685766844240000","716006292711030320000","530249260368885200000","132379072781930000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","84","1008","19152","536256","20913984","1087527168","72864320256","6120602901504","630422098854912","78172340258009088","11491334017927335936","1976509451083501780992"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"],["0","-2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187940,"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-ce4a47cd87c22637"]},{"id":187941,"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-ce4a47cd87c22637"]},{"id":187942,"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-ce4a47cd87c22637"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce4a47cd87c22637","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-702478000e7a23c3","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"9e5f342c6a71d0f7728fd6fe055cfe33cd1216e40e2f54fe23e00417d3a5c35e","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-3255c4dcfe77aa0b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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-bce60c664bc6aea1","kind":"fingerprint_related","node_a_id":"n-3255c4dcfe77aa0b","node_b_id":"n-702478000e7a23c3","fingerprint":"f6-7d88dc66b737aeee","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7d88dc66b737aeee"},"other_id":"n-3255c4dcfe77aa0b","other":{"id":"n-3255c4dcfe77aa0b","name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","first_index":0,"fingerprint":"f6-7d88dc66b737aeee","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","6","4"]],"card_rows":[["1"],["1","2"],["2","6","4"],["10","34","32","8"]],"row_sums_preview":["1","3","12","84","1008","19152","536256","20913984"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-54284d24a094e2d3","steps":["basis A013609"],"edges":["e-327e7f94e2e44f03"],"text":"A269944 \u2192 basis A013609"},"headline":"A269944 \u2192 basis A013609","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:   1;\n  1:   2,     1;\n  2:   4,     6,       2;\n  3:   8,    32,      34,       10;\n  4:  16,   144,     388,      360,       100;\n  5:  32,   560,    3224,     7316,      6320,       1700;\n  6:  64,  1952,   21008,    98456,    202856,     167720,       44200;\n  7: 128,  6272,  114240,   974208,   4048584,    7841112,     6294040,     1635400;\n  8: 256, 18944,  542080,  7660416,  56807568,  218111424,   404643680,   317972800,    81770000;\n  9: 512, 54528, 2315520, 50556032, 611542176, 4128714768, 14986529920, 26937784800, 20831772000, 5315050000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(n^2-2*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[(2 + (n^2 - 2*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] = (2 + (n^2 - 2*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-54284d24a094e2d3","steps":["basis A013609","reverse"],"edges":["e-327e7f94e2e44f03","e-ce4a47cd87c22637"],"text":"A269944 \u2192 basis A013609 \u2192 reverse"},"headline":"A269944 \u2192 basis A013609 \u2192 reverse","chain":null}