{"id":"n-f2085d6c0d79c3b9","revision":0,"first_index":0,"fingerprint":"f6-8e69101448399e5f","name":"shift_x_plus_one of basis_map_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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":true,"rows":[["1"],["15","11","2"],["225","330","181","44","4"],["3375","7425","6795","3311","906","132","8"],["50625","148500","190350","139260","63601","18568","3384","352","16"],["759375","2784375","4590000","4479750","2866575","1256651","382210","79640","10880","880","32"],["11390625","50118750","100996875","123255000","101455875","59341590","25289461","7912212","1803660","292160","31920","2112","64"],["170859375","877078125","2089040625","3060028125","3079636875","2252648475","1235011155","515550431","164668154","40047084","7299880","967120","88032","4928","128"],["2562890625","15035625000","41325187500","70634025000","84032943750","73785789000","49463574300","25823676120","10611099361","3443156816","879352432","174899648","26558560","2976512","232192","11264","256"],["38443359375","253726171875","790395468750","1544158687500","2120118806250","2172417266250","1721663181000","1079026037100","542154076335","220016797451","72287210178","19182685104","4080979392","686591136","89341632","8676096","592128","25344","512"],["576650390625","4228769531250","14723806640625","32364182812500","50368318593750","58995883237500","53961775256250","39468520080000","23444923915125","11421998875650","4588801077301","1522933183420","416798647380","93555010560","17054585760","2486081664","283002240","24245760","1470720","56320","1024"],["8649755859375","69774697265625","268526865234375","655882154296875","1140978403125000","1503718118718750","1559117981643750","1303599095493750","893751130119375","508161186361125","241363851621915","96164807352911","32181846882922","9033976646420","2118521197320","412001689440","65701103424","8448874368","854768640","65514240","3576320","123904","2048"],["129746337890625","1141767773437500","4812724160156250","12931577226562500","24866433474609375","36418298523750000","42209625836812500","39711720467925000","30864092965509375","20060878417717500","10997733084539850","5113796850856980","2023268287369681","681839580114264","195515254836264","47551711804960","9754577529840","1673448632064","237161354496","27282917376","2483838720","172226560","8546304","270336","4096"],["1946195068359375","18553726318359375","85009800585937500","249197159707031250","524869299931640625","845668400530078125","1083478538262656250","1132818288271312500","984209571303440625","719841639822215625","447363844794009000","237803773528228050","108596255839051695","42711138564494411","14479500778540226","4227622640501832","1060418002474688","227505505919120","41474510329824","6364915924224","811692380928","84471459072","6990364160","442517504","20127744","585728","8192"],["29192926025390625","299714040527343750","1483130388427734375","4710172654687500000","10784227856923828125","18956982626613281250","26604269079633984375","30601875046019062500","29391101817061359375","23889566458213706250","16597134852561388125","9927742175301951000","5139513035984301975","2310833439753440910","904207547565645301","308111125300458788","91369120639720924","23532425897012032","5245514225007056","1006704364329664","165138481540032","22925519924736","2657426274048","252474686464","19150337024","1115226112","46821376","1261568","16384"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","28","784","21952","614656","17210368","481890304","13492928512","377801998336","10578455953408","296196766695424","8293509467471872","232218265089212416","6502111422497947648","182059119829942534144"],"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":[["15","11","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188178,"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-86cf8bfc8a454a25"]},{"id":188179,"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-64ea7e1c16be0009","e-86cf8bfc8a454a25"]},{"id":188180,"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-64ea7e1c16be0009","e-86cf8bfc8a454a25"]},{"id":188181,"position":3,"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-64ea7e1c16be0009"]}],"region_exact":{},"incoming_edges":[{"id":"e-64ea7e1c16be0009","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-f2085d6c0d79c3b9","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":"504370cad474e83bac43e9dc9d01374efc072aa5dc62393843d2b50a0bc6192a","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-a19ad5e2037d7c6a","role":"sequence","name":"basis_map_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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-86cf8bfc8a454a25","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-f2085d6c0d79c3b9","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":"9deaa8a13ed6512888aee1c8d5f2b5c43c6f6757253faf7708cf19a474479119","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-a19ad5e2037d7c6a","role":"sequence","name":"basis_map_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","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-7ab1e22590a1cb7f","kind":"fingerprint_related","node_a_id":"n-f2085d6c0d79c3b9","node_b_id":"n-f03e5c22b4dd0de2","fingerprint":"f6-8e69101448399e5f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e69101448399e5f"},"other_id":"n-f03e5c22b4dd0de2","other":{"id":"n-f03e5c22b4dd0de2","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","first_index":0,"fingerprint":"f6-8e69101448399e5f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","33","45"],["36","396","1629","2970","2025"]],"card_rows":[["1"],["6","33","45"],["36","396","1629","2970","2025"],["216","3564","24462","89397","183465","200475","91125"]],"row_sums_preview":["1","84","7056","592704","49787136","4182119424","351298031616","29509034655744"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-eed4f744ce063213","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-bf76d3591eaf645b","e-fa8e415a5a123e3b"],"text":"A200536 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A200536 \u2192 basis A013610 \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:          15,           11,            2;\n  2:         225,          330,          181,            44,             4;\n  3:        3375,         7425,         6795,          3311,           906,           132,             8;\n  4:       50625,       148500,       190350,        139260,         63601,         18568,          3384,           352,           16;\n  5:      759375,      2784375,      4590000,       4479750,       2866575,       1256651,        382210,         79640,        10880,          880,          32;\n  6:    11390625,     50118750,    100996875,     123255000,     101455875,      59341590,      25289461,       7912212,      1803660,       292160,       31920,        2112,         64;\n  7:   170859375,    877078125,   2089040625,    3060028125,    3079636875,    2252648475,    1235011155,     515550431,    164668154,     40047084,     7299880,      967120,      88032,      4928,      128;\n  8:  2562890625,  15035625000,  41325187500,   70634025000,   84032943750,   73785789000,   49463574300,   25823676120,  10611099361,   3443156816,   879352432,   174899648,   26558560,   2976512,   232192,   11264,    256;\n  9: 38443359375, 253726171875, 790395468750, 1544158687500, 2120118806250, 2172417266250, 1721663181000, 1079026037100, 542154076335, 220016797451, 72287210178, 19182685104, 4080979392, 686591136, 89341632, 8676096, 592128, 25344, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (15+11*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(15 + 11*x + 2*x^2)*P[n - 1, 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] = (15 + 11*x + 2*x^2)*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-eed4f744ce063213","steps":["basis A007318","basis A007318"],"edges":["e-6e97878f95ac8f6d","e-64ea7e1c16be0009"],"text":"A200536 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A200536 \u2192 basis A007318 \u2192 basis A007318","chain":null}