{"id":"n-bdb6efcdc16252c4","revision":0,"first_index":1,"fingerprint":"f6-0d53f5dfcd557f8e","name":"shift_x_plus_one of Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","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":[["1"],["7","6"],["139","228","90"],["4609","11142","8694","2160"],["214297","684744","798876","399708","71280"],["12820999","50952270","79197516","59811912","21739428","2993760"],["937923283","4458531852","8662466646","8765879400","4841567964","1370228400","152681760"],["81109986433","448826839710","1046669680170","1329010724880","987810893820","426954893832","98362803600","9160905600"],["8094667157329","51108049132896","139090485294000","212565825877140","198847674546300","116052168719376","40989918282984","7928804037600","632102486400"],["915646445959303","6495808932145206","20209998782523816","36119577412673640","40759169905020240","30012962662763352","14352852961217304","4269416221548864","709401128349600","49303993939200"],["115769395774250971","911663193081238836","3191871057553772658","6532107338829618000","8635221054951288840","7684726211158603824","4646154310151007024","1875416621597119152","480386942879044320","69778844115494400","4289447472710400"],["16178897851297498561","140036724133338488502","544910209112388457278","1256548274464008630384","1904736354517964829960","1988710938727020701424","1455455865750931816128","744017958652702570272","259094112047104927536","58135244533854137280","7485488188617196800","411786957380198400"],["2476453714715956273993","23368550237650341034488","100048578860316829934628","256687350300714477996108","438934487823423806879448","526131718426497237237792","452333709555539942429712","280283658349648992924672","123790228489206857835792","37820760616270608236352","7535158988850911615040","869817131797568371200","43237630524920832000"],["412037876955756202893511","4209848227774411268784414","19666504615247810213799204","55565830733269639024864584","105816403271062486613457324","143225295518278981263300912","141513783881691565631404656","103123603923107362317184896","55273174138703883387346416","21444651668281787503265376","5825385922381632796724352","1041977270774784998810880","108846725323672505433600","4929089879840974848000"],["74042038979690777284629667","814316734563424712874129756","4121922461234323455753269166","12717273169800363437613905976","26691532072700373751432904676","40267449103590161796122677872","44966322627429318539364015264","37693261576879446079985783904","23782920775085956494834312624","11209503176797223916616866624","3871039234322396987599592736","945161196712824159027060480","153173976413361111625639680","14594505877197455560243200","606278055220439906304000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","457","26605","2168905","227515885","29189279305","4427906728045","775309695534025","153894138446140525","34147384417564148425","8375721854936609452525","2250323710125158451601225","657232272862733614702118125","207323118928441283115601335625"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-29","-30"],["18","18"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-15","-9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193346,"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-68d164b987bd9a85"]},{"id":193347,"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-68d164b987bd9a85"]},{"id":193348,"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-68d164b987bd9a85"]},{"id":193349,"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-68d164b987bd9a85"]}],"region_exact":{},"incoming_edges":[{"id":"e-68d164b987bd9a85","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-bdb6efcdc16252c4","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":"370b7b47066a61306825de0886baab8b9f3489d3fb797eb9840aa95ad3fc1e07","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-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","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":[],"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:               1;\n   2:               7,                6;\n   3:             139,              228,                90;\n   4:            4609,            11142,              8694,              2160;\n   5:          214297,           684744,            798876,            399708,             71280;\n   6:        12820999,         50952270,          79197516,          59811912,          21739428,           2993760;\n   7:       937923283,       4458531852,        8662466646,        8765879400,        4841567964,        1370228400,         152681760;\n   8:     81109986433,     448826839710,     1046669680170,     1329010724880,      987810893820,      426954893832,       98362803600,       9160905600;\n   9:   8094667157329,   51108049132896,   139090485294000,   212565825877140,   198847674546300,   116052168719376,    40989918282984,    7928804037600,    632102486400;\n  10: 915646445959303, 6495808932145206, 20209998782523816, 36119577412673640, 40759169905020240, 30012962662763352, 14352852961217304, 4269416221548864, 709401128349600, 49303993939200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18*n-29+(18*n-30)*x)*P(n-1,x) - (6+15*x+9*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(18*n - 29 + (18*n - 30)*x)*P[n - 1, x] - (6 + 15*x + 9*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] = 1; for(n = 2, N, my(i = n); v[i] = (18*n - 29 + (18*n - 30)*x)*v[i - 1] - (6 + 15*x + 9*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-7fdf5321df1ab451","steps":["x \u21a6 x+1"],"edges":["e-68d164b987bd9a85"],"text":"A290316 \u2192 x \u21a6 x+1"},"headline":"A290316 \u2192 x \u21a6 x+1","chain":null}