{"id":"n-a513e6b009fe2695","revision":0,"first_index":0,"fingerprint":"f6-72ea6b42576fa2d7","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 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":false,"rows":[["1"],["4","2"],["40","40","4"],["640","960","336","8"],["14080","28160","16800","2720","16"],["394240","985600","832000","262400","21824","32"],["13404160","40212480","43929600","20838400","3891840","174720","64"],["536166400","1876582400","2516828160","1600614400","479037440","56232960","1398016","128"],["24663654400","98654617600","156882288640","125355704320","52154798080","10480476160","801241600","11184640","256"],["1282510028800","5771295129600","10616094720000","10223620915200","5471716884480","1586595225600","222429388800","11329228800","89478144","512"],["74385581670400","371927908352000","776905113600000","876053004288000","574535968849920","219519606620160","46144922419200","4629855436800","159503984640","715827200","1024"],["4760677226905600","26183724747980800","61227028774912000","79143693877248000","61527770942668800","29296033441382400","8316685239582720","1301624802508800","95149172490240","2240214056960","5726621696","2048"],["333247405883392000","1999484435300352000","5174558603319705600","7544185693011968000","6787462157225164800","3879063259132723200","1396016568918343680","302579422136893440","35935526660014080","1938826874716160","31420263014400","45812981760","4096"],["25326802847137792000","164624218506395648000","467136692212885094400","758385403600515891200","775422430984378777600","517810017781363507200","226519446158573568000","63336329971551436800","10688736254860984320","976977884404121600","39279261702553600","440341812019200","366503870464","8192"],["2076797833465298944000","14537584834257092608000","44881760624375234560000","80301960900909203456000","91994396020325954355200","70349777059756113920000","36310395412362559488000","12500308829839294464000","2770197571584228065280","369394940918866903040","26265568751963340800","792630703043379200","6168450406973440","2932030996480","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","84","1944","61776","2496096","122451264","7066859904","469003965696","35185680999936","2944262181049344","271854634166360064","27454503364539273216","3010266776172085272576","356119638398631291469824"],"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":[["-2","-4"],["6","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-6","-6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191211,"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-38143c045283e3d7"]},{"id":191212,"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-38143c045283e3d7"]},{"id":191213,"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-38143c045283e3d7"]},{"id":191214,"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-38143c045283e3d7"]}],"region_exact":{},"incoming_edges":[{"id":"e-38143c045283e3d7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a513e6b009fe2695","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":"5128733a28f564f1ddbc5c3f3c288016a124f175f6345dbee4089a912e868181","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-c060a08f0b5770ad","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 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":[{"id":"l-431df6e7b0f167d9","kind":"fingerprint_related","node_a_id":"n-0b7c6f7d1239dc9e","node_b_id":"n-a513e6b009fe2695","fingerprint":"f6-72ea6b42576fa2d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ea6b42576fa2d7"},"other_id":"n-0b7c6f7d1239dc9e","other":{"id":"n-0b7c6f7d1239dc9e","name":"shift_x_plus_one of Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","first_index":0,"fingerprint":"f6-72ea6b42576fa2d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["10","10","1"]],"card_rows":[["1"],["2","1"],["10","10","1"],["80","120","42","1"]],"row_sums_preview":["1","3","21","243","3861","78003","1913301","55209843"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ddb0cd3a4d6edf09","steps":["x \u21a6 x+1"],"edges":["e-b96a22dcbe3b84b0"],"text":"A142458 \u2192 x \u21a6 x+1"},"headline":"A142458 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-707296e276059675","kind":"fingerprint_related","node_a_id":"n-a513e6b009fe2695","node_b_id":"n-78dc37aeb5f3a608","fingerprint":"f6-72ea6b42576fa2d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ea6b42576fa2d7"},"other_id":"n-78dc37aeb5f3a608","other":{"id":"n-78dc37aeb5f3a608","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.","first_index":0,"fingerprint":"f6-72ea6b42576fa2d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["4","40","40"]],"card_rows":[["1"],["2","4"],["4","40","40"],["8","336","960","640"]],"row_sums_preview":["1","6","84","1944","61776","2496096","122451264","7066859904"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c060a08f0b5770ad","steps":["x \u21a6 x/(x+1)"],"edges":["e-49dd9e1b36bd93a9"],"text":"A257616 \u2192 x \u21a6 x/(x+1)"},"headline":"A257616 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-871dba154d87e591","kind":"fingerprint_related","node_a_id":"n-7e2941a6c64b355d","node_b_id":"n-a513e6b009fe2695","fingerprint":"f6-72ea6b42576fa2d7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-72ea6b42576fa2d7"},"other_id":"n-7e2941a6c64b355d","other":{"id":"n-7e2941a6c64b355d","name":"mobius_x_over_x_plus_one of Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).","first_index":0,"fingerprint":"f6-72ea6b42576fa2d7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","10","10"]],"card_rows":[["1"],["1","2"],["1","10","10"],["1","42","120","80"]],"row_sums_preview":["1","3","21","243","3861","78003","1913301","55209843"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ddb0cd3a4d6edf09","steps":["x \u21a6 x/(x+1)"],"edges":["e-f05af37e7e72e4b4"],"text":"A142458 \u2192 x \u21a6 x/(x+1)"},"headline":"A142458 \u2192 x \u21a6 x/(x+1)","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:             4,             2;\n  2:            40,            40,              4;\n  3:           640,           960,            336,              8;\n  4:         14080,         28160,          16800,           2720,            16;\n  5:        394240,        985600,         832000,         262400,         21824,            32;\n  6:      13404160,      40212480,       43929600,       20838400,       3891840,        174720,           64;\n  7:     536166400,    1876582400,     2516828160,     1600614400,     479037440,      56232960,      1398016,         128;\n  8:   24663654400,   98654617600,   156882288640,   125355704320,   52154798080,   10480476160,    801241600,    11184640,      256;\n  9: 1282510028800, 5771295129600, 10616094720000, 10223620915200, 5471716884480, 1586595225600, 222429388800, 11329228800, 89478144, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6*n-2+(6*n-4)*x)*P(n-1,x) - (6*x+6*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[(6*n - 2 + (6*n - 4)*x)*P[n - 1, x] - (6*x + 6*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] = (6*n - 2 + (6*n - 4)*x)*v[i - 1] - (6*x + 6*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-c060a08f0b5770ad","steps":["x \u21a6 x+1"],"edges":["e-38143c045283e3d7"],"text":"A257616 \u2192 x \u21a6 x+1"},"headline":"A257616 \u2192 x \u21a6 x+1","chain":null}