{"id":"n-47ad3360622db9f1","revision":0,"first_index":0,"fingerprint":"f6-b4bba0102ede147f","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.","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"],["6","3"],["60","60","9"],["840","1260","474","27"],["15120","30240","18600","3480","81"],["332640","831600","714000","239400","24846","243"],["8648640","25945920","28652400","14061600","2881860","175380","729"],["259459200","908107200","1227108960","797504400","251270040","33454260","1232034","2187"],["8821612800","35286451200","56423727360","45768602880","19675444320","4237410240","380317200","8637360","6561"],["335221286400","1508495788800","2786591808000","2713424313600","1485537641280","450106826400","68885594400","4269862800","60500886","19683"],["14079294028800","70396470144000","147541474080000","167787075456000","111886435140480","44069715930240","9791970400800","1093061995200","47573499660","423624300","59049"],["647647525324800","3562061389286400","8352038350656000","10868712158304000","8557992393350400","4166746362979200","1229026815213120","205722553528800","17061958923240","527544739260","2965724394","177147"],["32382376266240000","194294257597440000","503954250466867200","738740557691136000","671376666914553600","390300155083238400","144456848995979520","32793512927535360","4215843777665520","263315010198240","5832649375800","20761133640","531441"],["1748648318376960000","11366214069450240000","32313725628555571200","52697136193103001600","54323195092707340800","36768388614839481600","16433010277459776000","4752966259007366400","846999360441954720","84840701959213200","4031382244234800","64366754470200","145331124126","1594323"],["101421602465863680000","709951217261045760000","2195363198969740800000","3942813369424849920000","4547551050197137382400","3515803375979486592000","1845615950953014240000","652012306371474048000","150352072364689715520","21347480733267648960","1684538217621277200","61371868226104800","709487610413460","1017327434820","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","129","2601","67521","2142729","80366529","3478138281","170602209921","9352593642249","566693494358529","37607540018206761","2712783638141894721","211339220410232328969","17683978245311359800129"],"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","-1"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189767,"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-cbc7ba973fb794f3"]},{"id":189768,"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-cbc7ba973fb794f3"]},{"id":189769,"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-cbc7ba973fb794f3"]},{"id":189770,"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-cbc7ba973fb794f3"]}],"region_exact":{},"incoming_edges":[{"id":"e-cbc7ba973fb794f3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-47ad3360622db9f1","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":"758d38859a5251c5c7c7656ab6336052820d7f1618ca923358c484828571d20f","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-930be05ae3b321d9","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.","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-cda2e0719df0a688","kind":"fingerprint_related","node_a_id":"n-47ad3360622db9f1","node_b_id":"n-3447ba8ca39d0e59","fingerprint":"f6-b4bba0102ede147f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b4bba0102ede147f"},"other_id":"n-3447ba8ca39d0e59","other":{"id":"n-3447ba8ca39d0e59","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.","first_index":0,"fingerprint":"f6-b4bba0102ede147f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","6"],["9","60","60"]],"card_rows":[["1"],["3","6"],["9","60","60"],["27","474","1260","840"]],"row_sums_preview":["1","9","129","2601","67521","2142729","80366529","3478138281"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-930be05ae3b321d9","steps":["x \u21a6 x/(x+1)"],"edges":["e-e095ae63548ad51f"],"text":"A257621 \u2192 x \u21a6 x/(x+1)"},"headline":"A257621 \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:            6,             3;\n  2:           60,            60,             9;\n  3:          840,          1260,           474,            27;\n  4:        15120,         30240,         18600,          3480,            81;\n  5:       332640,        831600,        714000,        239400,         24846,          243;\n  6:      8648640,      25945920,      28652400,      14061600,       2881860,       175380,         729;\n  7:    259459200,     908107200,    1227108960,     797504400,     251270040,     33454260,     1232034,       2187;\n  8:   8821612800,   35286451200,   56423727360,   45768602880,   19675444320,   4237410240,   380317200,    8637360,     6561;\n  9: 335221286400, 1508495788800, 2786591808000, 2713424313600, 1485537641280, 450106826400, 68885594400, 4269862800, 60500886, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+2+(4*n-1)*x)*P(n-1,x) - (4*x+4*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[(4*n + 2 + (4*n - 1)*x)*P[n - 1, x] - (4*x + 4*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] = (4*n + 2 + (4*n - 1)*x)*v[i - 1] - (4*x + 4*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-930be05ae3b321d9","steps":["x \u21a6 x+1"],"edges":["e-cbc7ba973fb794f3"],"text":"A257621 \u2192 x \u21a6 x+1"},"headline":"A257621 \u2192 x \u21a6 x+1","chain":null}