{"id":"n-59c538f64c72051a","revision":0,"first_index":0,"fingerprint":"f6-d0b4c8550cf1eaf3","name":"shift_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","4","1"],["24","48","34","10","1"],["192","576","688","416","132","20","1"],["1920","7680","12960","12000","6616","2192","414","38","1"],["23040","115200","251520","314880","249184","129312","44040","9520","1200","72","1"],["322560","1935360","5187840","8198400","8481984","6026496","2996176","1040112","246360","37792","3378","138","1"],["5160960","36126720","114831360","219340800","280691712","253854720","166667008","80253952","28254016","7150528","1256080","143584","9436","268","1"],["92897280","743178240","2732728320","6123479040","9344563200","10276761600","8403141120","5192901120","2439304576","868063744","231018592","44840992","6080200","534448","26422","526","1"],["1857945600","16721510400","69827788800","179601408000","318459617280","412765839360","404516444160","305735086080","180085287424","82936793600","29766314880","8240939520","1727332320","265839264","28586280","1974000","74520","1040","1"],["40874803200","408748032000","1910897049600","5548754534400","11213150699520","16743681884160","19141204684800","17123964211200","12148192134144","6881932806144","3118649108224","1127031084800","322128745920","71773488384","12185647824","1522071024","132316200","7287360","212106","2066","1"],["980995276800","10790948044800","55834981171200","180666630144000","409923863838720","693109950382080","905884792750080","936809874063360","778154784202752","523990938476544","287393951881216","128498343358464","46702482123264","13703793838592","3209283563904","589426776576","82718934144","8530287744","608024208","26999456","609444","4116","1"],["25505877196800","306070526361600","1736525139148800","6195802669056000","15597162729308160","29460350907187200","43343568874045440","50894241378140160","48475455918489600","37856600313692160","24400495182336000","13022957645967360","5757001489079296","2102409905780736","630642880775424","153935932663040","30155518800000","4647171910656","547361433696","47209587744","2788593720","100618640","1766574","8214","1"],["714164561510400","9284139299635200","57311706061209600","223533507752755200","618302847555993600","1290480484220928000","2111070502866124800","2775407739155251200","2982954762351083520","2651975113661153280","1965701719915315200","1220654244906270720","636471628964737024","278638250680373248","102172348133180416","31227475811796992","7894001141970944","1632098370366464","271667949641088","35610556540416","3562018972512","259540115744","12807898376","377594672","5160064","16408","1"],["21424936845312000","299949115834368000","2001446183632896000","8471777110917120000","25541530162141593600","58379739162948403200","105129934398436147200","153017773588257177600","183233118103537582080","182786408542273536000","153240435979997184000","108609030722949120000","65310806645193007104","33374453500516564992","14488269392303761408","5331348383389751296","1655954757377170432","431353325395865600","93364496715666176","16579435150623488","2373780800504256","267521895044352","22938041420112","1423120195952","59037879992","1427489504","15168418","32794","1"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","9","117","2025","43821","1137969","34476597","1193741145","46499519421","2012538782529","95814830803077","4976336924751465","279994127521947021","16965737848665595089","1101437363668150585557"],"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","0","-1"],["2","4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-3","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185414,"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-f8c3616db5beeda3"]},{"id":185415,"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-f8c3616db5beeda3"]},{"id":185416,"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-f8c3616db5beeda3"]}],"region_exact":{},"incoming_edges":[{"id":"e-f8c3616db5beeda3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-59c538f64c72051a","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":"df728ac25ae3436f956130ce4e073e46a68fdddb20ef078df248a7681a51de74","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-b3367e055aead099","role":"sequence","name":"Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","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-e55d91b5e2563727","kind":"fingerprint_related","node_a_id":"n-59c538f64c72051a","node_b_id":"n-3e49d29390a1ef28","fingerprint":"f6-d0b4c8550cf1eaf3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d0b4c8550cf1eaf3"},"other_id":"n-3e49d29390a1ef28","other":{"id":"n-3e49d29390a1ef28","name":"mobius_x_over_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","first_index":0,"fingerprint":"f6-d0b4c8550cf1eaf3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4","4"],["1","10","34","48","24"]],"card_rows":[["1"],["1","4","4"],["1","10","34","48","24"],["1","20","132","416","688","576","192"]],"row_sums_preview":["1","9","117","2025","43821","1137969","34476597","1193741145"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":null,"headline":"mobius_x_over_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","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,           4,           1;\n  2:         24,          48,          34,           10,            1;\n  3:        192,         576,         688,          416,          132,           20,            1;\n  4:       1920,        7680,       12960,        12000,         6616,         2192,          414,           38,            1;\n  5:      23040,      115200,      251520,       314880,       249184,       129312,        44040,         9520,         1200,          72,           1;\n  6:     322560,     1935360,     5187840,      8198400,      8481984,      6026496,      2996176,      1040112,       246360,       37792,        3378,        138,          1;\n  7:    5160960,    36126720,   114831360,    219340800,    280691712,    253854720,    166667008,     80253952,     28254016,     7150528,     1256080,     143584,       9436,       268,        1;\n  8:   92897280,   743178240,  2732728320,   6123479040,   9344563200,  10276761600,   8403141120,   5192901120,   2439304576,   868063744,   231018592,   44840992,    6080200,    534448,    26422,     526,     1;\n  9: 1857945600, 16721510400, 69827788800, 179601408000, 318459617280, 412765839360, 404516444160, 305735086080, 180085287424, 82936793600, 29766314880, 8240939520, 1727332320, 265839264, 28586280, 1974000, 74520, 1040, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+2+4*n*x+(2*n-1)*x^2)*P(n-1,x) - (2*x+3*x^2+x^3)*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[(2*n + 2 + 4*n*x + (2*n - 1)*x^2)*P[n - 1, x] - (2*x + 3*x^2 + x^3)*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] = (2*n + 2 + 4*n*x + (2*n - 1)*x^2)*v[i - 1] - (2*x + 3*x^2 + x^3)*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":null,"headline":"shift_x_plus_one of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.","chain":null}