{"id":"n-fb5f5aabac06a450","revision":0,"first_index":0,"fingerprint":"f6-2df49debe483a37b","name":"reversal of Scaled array A078740 ((3,2)-Stirling2).","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"],["2","3","1"],["5","15","16","7","1"],["14","63","114","105","51","12","1"],["42","252","644","910","771","396","118","18","1"],["132","990","3270","6240","7580","6083","3235","1110","230","25","1"],["429","3861","15642","37620","59625","65331","50464","27483","10365","2600","402","33","1"],["1430","15015","72072","209209","409332","569415","578690","434493","240751","97188","27825","5390","651","42","1"],["4862","58344","323752","1101100","2564562","4328324","5462072","5241060","3849715","2161288","918484","289884","65828","10220","996","52","1"],["16796","226746","1428102","5569200","15049008","29875482","45072846","52723476","48344922","34895685","19793907","8757108","2979312","762048","141552","18096","1458","63","1"],["58786","881790","6214520","27335490","84075030","191962368","337119120","465527790","512370495","452841675","322239204","184309005","84223665","30417390","8536920","1816248","282375","30345","2060","75","1"],["208012","3432198","26761196","131070170","452119250","1167203136","2339265654","3725785008","4788113220","5012613375","4297304088","3021922137","1740293049","816717990","309674475","93647466","22169994","4001613","530090","48675","2827","88","1"],["742900","13372200","114287736","616815012","2358346386","6792767520","15300852380","27618023268","40604633586","49159626648","49352002920","41238216630","28710585099","16629143688","7979929254","3150785550","1013169465","261545328","53136116","8264718","946275","75240","3786","102","1"],["2674440","52151580","484816540","2858679200","11999499096","38148880770","95416220042","192514458708","318754907900","438294903618","504554375754","488794023744","399589566168","275827551795","160539021279","78511523610","32078801898","10860320757","3012028305","673878348","119118428","16168867","1619007","112710","4966","117","1"],["9694845","203591745","2045277990","13081354650","59805790135","208020580495","572056089332","1275982857006","2349839462971","3618158641097","4699678925942","5181437896908","4867860327057","3904937056923","2675584398912","1563800707875","777335641173","327060707064","115687505934","34086125073","8263934679","1622156250","252317156","30204083","2671123","164346","6398","133","1"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","6","44","360","3152","28896","273856","2661504","26380544","265655808","2710244352","27952883712","290977271808","3053105307648","32256844087296"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","0","-2"],["4","9","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183973,"position":0,"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-eceef4cd5dcdaaee"]},{"id":183974,"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-eceef4cd5dcdaaee"]},{"id":183975,"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-eceef4cd5dcdaaee"]}],"region_exact":{},"incoming_edges":[{"id":"e-eceef4cd5dcdaaee","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":2}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-fb5f5aabac06a450","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"b1f7d3145e8159c31c06c15f7646a30a3d33a0a70b52c33a2ba455184b294c1c","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-4e0759372c05411f","role":"sequence","name":"Scaled array A078740 ((3,2)-Stirling2).","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-3fc3e56f43c60345","kind":"fingerprint_related","node_a_id":"n-4e0759372c05411f","node_b_id":"n-fb5f5aabac06a450","fingerprint":"f6-2df49debe483a37b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2df49debe483a37b"},"other_id":"n-4e0759372c05411f","other":{"id":"n-4e0759372c05411f","name":"Scaled array A078740 ((3,2)-Stirling2).","first_index":0,"fingerprint":"f6-2df49debe483a37b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3","2"],["1","7","16","15","5"]],"card_rows":[["1"],["1","3","2"],["1","7","16","15","5"],["1","12","51","105","114","63","14"]],"row_sums_preview":["1","6","44","360","3152","28896","273856","2661504"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A090452"],"derivation":{"root":"n-4e0759372c05411f","steps":[],"edges":[],"text":"A090452"},"headline":"A090452: Scaled array A078740 ((3,2)-Stirling2).","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:     2,      3,       1;\n  2:     5,     15,      16,       7,        1;\n  3:    14,     63,     114,     105,       51,       12,        1;\n  4:    42,    252,     644,     910,      771,      396,      118,       18,        1;\n  5:   132,    990,    3270,    6240,     7580,     6083,     3235,     1110,      230,       25,        1;\n  6:   429,   3861,   15642,   37620,    59625,    65331,    50464,    27483,    10365,     2600,      402,      33,       1;\n  7:  1430,  15015,   72072,  209209,   409332,   569415,   578690,   434493,   240751,    97188,    27825,    5390,     651,     42,      1;\n  8:  4862,  58344,  323752, 1101100,  2564562,  4328324,  5462072,  5241060,  3849715,  2161288,   918484,  289884,   65828,  10220,    996,    52,    1;\n  9: 16796, 226746, 1428102, 5569200, 15049008, 29875482, 45072846, 52723476, 48344922, 34895685, 19793907, 8757108, 2979312, 762048, 141552, 18096, 1458, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (4*n+2+9*n*x+(5*n-2)*x^2)*P(n-1,x) - (2*x+4*x^2+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[Together[((4*n + 2 + 9*n*x + (5*n - 2)*x^2)*P[n - 1, x] - (2*x + 4*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2)]];\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] = ((4*n + 2 + 9*n*x + (5*n - 2)*x^2)*v[i - 1] - (2*x + 4*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2)); 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-4e0759372c05411f","steps":["reverse"],"edges":["e-eceef4cd5dcdaaee"],"text":"A090452 \u2192 reverse"},"headline":"A090452 \u2192 reverse","chain":null}
