{"id":"n-c781d765aa7ea874","revision":0,"first_index":0,"fingerprint":"f6-5bb7eea759246da4","name":"shift_x_plus_one of Triangle of 3-Eulerian 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"],["4","3"],["20","28","9"],["120","240","148","27"],["840","2160","1940","700","81"],["6720","21000","24600","13200","3124","243"],["60480","221760","319200","226800","81620","13468","729"],["604800","2540160","4327680","3822000","1850520","475440","56788","2187"],["6652800","31449600","61810560","65197440","39707640","13955760","2661140","235900","6561"],["79833600","419126400","932601600","1143072000","839240640","375643800","99701400","14485200","969844","19683"],["1037836800","5987520000","14869008000","20756736000","17840813760","9719075520","3327416400","685119600","77275220","3958108","59049"],["14529715200","91329638400","250278336000","391883184000","385975497600","248039003520","104371935360","28087752000","4574368920","406166640","16068628","177147"],["217945728000","1482030950400","4440903667200","7705938240000","8556445497600","6340145011200","3171698449920","1059677498880","228634846440","29883213360","2111176340","64983100","531441"],["3487131648000","25499650176000","82921084646400","157908944793600","195148604851200","163862495596800","95118434611200","38082383539200","10306498762560","1809626418600","191966338200","10880797200","262058164","1594323"],["59281238016000","463788509184000","1626311022336000","3371620412160000","4589748145382400","4308133205376000","2848790232288000","1332008181504000","435499940715840","96929001449280","14011149297600","1217082812400","55714276820","1054609948","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","57","535","5721","68887","924057","13679575","221677401","3904694167","74304818457","1519491843415","33235479793881","774347965831447","19147394894191257"],"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":{"terms":[{"coeff":{"coeffs":[["3","2"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182701,"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-39ef0d1884075014"]},{"id":182702,"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-39ef0d1884075014"]},{"id":182703,"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-39ef0d1884075014"]},{"id":182704,"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-39ef0d1884075014"]}],"region_exact":{},"incoming_edges":[{"id":"e-39ef0d1884075014","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c781d765aa7ea874","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":"312bc21b2dd3a8af559b027f505609677e8fa9438645942bd87eb91d7e42319c","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-4c402d5a817396fa","role":"sequence","name":"Triangle of 3-Eulerian 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:        4,         3;\n  2:       20,        28,         9;\n  3:      120,       240,       148,         27;\n  4:      840,      2160,      1940,        700,        81;\n  5:     6720,     21000,     24600,      13200,      3124,       243;\n  6:    60480,    221760,    319200,     226800,     81620,     13468,      729;\n  7:   604800,   2540160,   4327680,    3822000,   1850520,    475440,    56788,     2187;\n  8:  6652800,  31449600,  61810560,   65197440,  39707640,  13955760,  2661140,   235900,   6561;\n  9: 79833600, 419126400, 932601600, 1143072000, 839240640, 375643800, 99701400, 14485200, 969844, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+3+(n+2)*x)*P(n-1,x) - (x+x^2)*d/dx 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 + 3 + (n + 2)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], 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 + 3 + (n + 2)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); 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-4c402d5a817396fa","steps":["x \u21a6 x+1"],"edges":["e-39ef0d1884075014"],"text":"A144697 \u2192 x \u21a6 x+1"},"headline":"A144697 \u2192 x \u21a6 x+1","chain":null}
