{"id":"n-3b96364a23191cc0","revision":0,"first_index":0,"fingerprint":"f6-5e64a37ca1ebb259","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["961","682","121"],["29791","31713","11253","1331"],["923521","1310804","697686","165044","14641"],["28629151","50793655","36047110","12790910","2269355","161051"],["887503681","1889523966","1676190615","793036420","211050015","29955486","1771561"],["27512614111","68337783437","72746672691","43022225785","15265951085","3250170231","384428737","19487171"],["852891037441","2421110041768","3006862471228","2133902398936","946488967270","268680739096","47669163388","4832818408","214358881"],["26439622160671","84436212706659","119844947067516","99226461550524","52814084373666","18740481551946","4433232195084","674178167916","59806127799","2357947691"],["819628286980801","2908358437673810","4643991698866245","4394314725808920","2728727692639410","1161909856220652","343575495119010","69665077351320","9269949808845","730963784210","25937424601"],["25408476896404831","99175022724676921","175955685479265505","187307665187605215","132928020455719830","66035210161873722","23431848767116482","5938947844200030","1053684294938715","124629325207805","8844661788941","285311670611"],["787662783788549761","3353918950325437692","6545551499828676786","7742050161087682220","6181152951190972095","3509299740031003512","1452774623561221884","441857719608482232","97992639429300495","15454036325767820","1645107092743026","106135941467292","3138428376721"],["24417546297445042591","112635778081762615823","239805204948268794978","312004621491833593466","276778293258884639365","176780974404061801917","83638310470738917036","29678110167036389916","7898206738001619897","1556994159821107865","220992719458479826","21386392205659338","1264786635818563","34522712143931"],["756943935220796320321","3760302129806536559014","8672954912295721418371","12310000520677798142204","12012177927435593348441","8524771432373646892442","4537378343037586249203","1840042830356256174792","571303620715450505883","135147093072472162682","23977710061245061121","3093898072418717564","274458699972628171","14982857070466054","379749833583241"],["23465261991844685929951","124895749311431392852965","310224925709039266118655","477012520176264678010405","507787521477959173365915","396401871605374580498553","234431214390275289542155","106952489514457390159785","37950883376097783605085","10473899713116592607855","2229927035695790684253","359665650918675916815","42541098495757366505","3483514268883357555","176583672616207065","4177248169415651"],["727423121747185263828481","4129886110564664723671376","10990825939405962571060920","18199862308263636945627760","20988550887755645832457820","17874120756024162902480208","11627788233757654361290888","5894270533241207279917040","2352954769318062583515270","742150608243689990499440","184340634950852029898248","35678832571132650948048","5275096213473913446620","575941025788715115760","43792750808819352120","2071915092030162896","45949729863572161"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1764","74088","3111696","130691232","5489031744","230539333248","9682651996416","406671383849472","17080198121677824","717368321110468608","30129469486639681536","1265437718438866624512","53148384174432398229504","2232232135326160725639168","93753749683698750476845056"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["961","682","121"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["31","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194492,"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-5da8f9342dab5fdd"]},{"id":194493,"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-5da8f9342dab5fdd","e-acff801d655293e5"]},{"id":194494,"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-5da8f9342dab5fdd","e-acff801d655293e5"]},{"id":194495,"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-5da8f9342dab5fdd"]},{"id":194496,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-acff801d655293e5"]}],"region_exact":{},"incoming_edges":[{"id":"e-acff801d655293e5","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3b96364a23191cc0","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"a4284250cd78c713d2303a6e91e1251d81352dabdbbd7ea1c62204ec4f5c5bfe","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c02d1829f7ed6fe0","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","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]"}]},{"id":"e-5da8f9342dab5fdd","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3b96364a23191cc0","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":"b8580f5208f32b1865bb2aa40813af9ec1511486d352f81b27a61c1b7f990271","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-c02d1829f7ed6fe0","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.","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-cbf626f6971e6b89","kind":"fingerprint_related","node_a_id":"n-aad6f7f53daa6d5b","node_b_id":"n-3b96364a23191cc0","fingerprint":"f6-5e64a37ca1ebb259","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5e64a37ca1ebb259"},"other_id":"n-aad6f7f53daa6d5b","other":{"id":"n-aad6f7f53daa6d5b","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.","first_index":0,"fingerprint":"f6-5e64a37ca1ebb259","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","31"],["121","682","961"]],"card_rows":[["1"],["11","31"],["121","682","961"],["1331","11253","31713","29791"]],"row_sums_preview":["1","42","1764","74088","3111696","130691232","5489031744","230539333248"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6ebe74518d42cbe","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-17e7dceca4b24148","e-55b012587d4839e6"],"text":"A038259 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A038259 \u2192 basis A013611 \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:               961,               682,                121;\n  1:             29791,             31713,              11253,               1331;\n  2:            923521,           1310804,             697686,             165044,              14641;\n  3:          28629151,          50793655,           36047110,           12790910,            2269355,            161051;\n  4:         887503681,        1889523966,         1676190615,          793036420,          211050015,          29955486,           1771561;\n  5:       27512614111,       68337783437,        72746672691,        43022225785,        15265951085,        3250170231,         384428737,         19487171;\n  6:      852891037441,     2421110041768,      3006862471228,      2133902398936,       946488967270,      268680739096,       47669163388,       4832818408,        214358881;\n  7:    26439622160671,    84436212706659,    119844947067516,     99226461550524,     52814084373666,    18740481551946,     4433232195084,     674178167916,      59806127799,      2357947691;\n  8:   819628286980801,  2908358437673810,   4643991698866245,   4394314725808920,   2728727692639410,  1161909856220652,   343575495119010,   69665077351320,    9269949808845,    730963784210,   25937424601;\n  9: 25408476896404831, 99175022724676921, 175955685479265505, 187307665187605215, 132928020455719830, 66035210161873722, 23431848767116482, 5938947844200030, 1053684294938715, 124629325207805, 8844661788941, 285311670611;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (31+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 961 + 682*x + 121*x^2.","mathematica":"P[0, x_] := 961 + 682*x + 121*x^2;\nP[n_, x_] := P[n, x] = Expand[(31 + 11*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 961 + 682*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (31 + 11*x)*v[i - 1]); 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-cfc5b48d32898a3e","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-1af17fa5254d2e99","e-5da8f9342dab5fdd"],"text":"A038301 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A038301 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}