{"id":"n-e23fac9b1096b64b","revision":0,"first_index":1,"fingerprint":"f6-09455fa73ad50cf5","name":"shift_x_plus_one of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["42","42","9"],["336","504","222","27"],["3024","6048","4074","1050","81"],["30240","75600","68880","27720","4686","243"],["332640","997920","1149120","635040","171402","20202","729"],["3991680","13970880","19474560","13759200","5181456","998424","85182","2187"],["51891840","207567360","339958080","293388480","142947504","39076128","5588394","353850","6561"],["726485760","3269185920","6155170560","6286896000","3776582880","1352317680","279163920","30418920","1454766","19683"],["10897286400","54486432000","115978262400","136994457600","98124475680","43735839840","11978699040","1918334880","162277962","5937162","59049"],["174356582400","958961203200","2277532857600","3056688835200","2547438284160","1364214519360","469673709120","101115907200","12808232976","852949944","24102942","177147"],["2964061900800","17784371404800","46629488505600","70124037984000","66740274881280","41844957073920","17444341474560","4768548744960","823085447184","83672997408","4433470314","97474650","531441"],["53353114214400","346795242393600","995053015756800","1658043920332800","1775852304145920","1278127465655040","627781668433920","209453109465600","46379244431520","6514655106960","537505746960","22849674120","393087246","1594323"],["1013709170073600","7095964190515200","22117829903769600","40459444945920000","48192355526515200","39204012168921600","22220563811846400","8791253997926400","2395249673937120","436180506521760","50440137471360","3407831874720","116999981322","1581914922","4782969"],["20274183401472000","152056375511040000","511656365316096000","1019578012637184000","1337509917667123200","1214560804378521600","781536039392908800","358712503903353600","116655536066509440","26355232637173440","3993845426887680","383751449172000","21382991098896","596073311064","6356357502","14348907"],["425757851430912000","3406062811447296000","12306429324693504000","26538906072526848000","38031338789898854400","38158111717396070400","27512331047814412800","14400407527361856000","5462359511801512320","1482818152311175680","281129393430325440","35788277906821440","2878707064094064","133066533081888","3024861057834","25511523450","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["93","1089","14277","207369","3307053","57463569","1080778197","21877696089","474282062013","10963667361249","269211371890917","6997914490039209","191980530451972173","5543294553222558129","168044455302032392437"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["42","42","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","3"],["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":189816,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-dbbb61dc8289264e"]},{"id":189817,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-dbbb61dc8289264e"]},{"id":189818,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-dbbb61dc8289264e"]},{"id":189819,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-dbbb61dc8289264e"]}],"region_exact":{},"incoming_edges":[{"id":"e-dbbb61dc8289264e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-e23fac9b1096b64b","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":"87186a60ffa75b770b9066a40e8f9590a01f4407995bd429665ae1f560db64ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b58ba5723c3a3a3f","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-7820bed07b7c3685","kind":"fingerprint_related","node_a_id":"n-e23fac9b1096b64b","node_b_id":"n-1bebae31ffd0b112","fingerprint":"f6-09455fa73ad50cf5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-09455fa73ad50cf5"},"other_id":"n-1bebae31ffd0b112","other":{"id":"n-1bebae31ffd0b112","name":"mobius_x_over_x_plus_one of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","first_index":1,"fingerprint":"f6-09455fa73ad50cf5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["9","42","42"],["27","222","504","336"],["81","1050","4074","6048","3024"]],"card_rows":[["9","42","42"],["27","222","504","336"],["81","1050","4074","6048","3024"],["243","4686","27720","68880","75600","30240"]],"row_sums_preview":["93","1089","14277","207369","3307053","57463569","1080778197","21877696089"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b58ba5723c3a3a3f","steps":["x \u21a6 x/(x+1)"],"edges":["e-60089487a9b4d630"],"text":"A257180 \u2192 x \u21a6 x/(x+1)"},"headline":"A257180 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:           42,           42,             9;\n   2:          336,          504,           222,            27;\n   3:         3024,         6048,          4074,          1050,            81;\n   4:        30240,        75600,         68880,         27720,          4686,           243;\n   5:       332640,       997920,       1149120,        635040,        171402,         20202,          729;\n   6:      3991680,     13970880,      19474560,      13759200,       5181456,        998424,        85182,         2187;\n   7:     51891840,    207567360,     339958080,     293388480,     142947504,      39076128,      5588394,       353850,        6561;\n   8:    726485760,   3269185920,    6155170560,    6286896000,    3776582880,    1352317680,    279163920,     30418920,     1454766,     19683;\n   9:  10897286400,  54486432000,  115978262400,  136994457600,   98124475680,   43735839840,  11978699040,   1918334880,   162277962,   5937162,    59049;\n  10: 174356582400, 958961203200, 2277532857600, 3056688835200, 2547438284160, 1364214519360, 469673709120, 101115907200, 12808232976, 852949944, 24102942, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+6+(n+3)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 42 + 42*x + 9*x^2.","mathematica":"P[1, x_] := 42 + 42*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(n + 6 + (n + 3)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 42 + 42*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = (n + 6 + (n + 3)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b58ba5723c3a3a3f","steps":["x \u21a6 x+1"],"edges":["e-dbbb61dc8289264e"],"text":"A257180 \u2192 x \u21a6 x+1"},"headline":"A257180 \u2192 x \u21a6 x+1","chain":null}