{"id":"n-0c5ba5f813be2bea","revision":1,"first_index":0,"fingerprint":"f6-078627b5c58f7a3e","name":"shift_x_plus_one of Triangle of B-analogs of Stirling numbers of 2nd kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["6","6","1"],["24","38","16","1"],["116","256","184","44","1"],["648","1844","1896","824","126","1"],["4088","14296","19236","12264","3606","370","1"],["28640","119160","199088","168804","74936","15798","1100","1"],["219920","1063168","2128960","2265696","1361256","448160","70176","3288","1"],["1832224","10107152","23642944","30380608","23163568","10512872","2670032","317984","9850","1"],["16430176","101954912","273253264","412380480","382473456","222758864","79483944","15988752","1471702","29534","1"],["157554048","1087209056","3288406144","5709322640","6252959296","4463865328","2067537088","595875752","96672632","6945014","88584","1"],["1606879040","12214299648","41189952640","80967914048","102443505968","86839405696","49687988736","18812642848","4464490460","591596576","33307704","265732","1"],["17350255744","144126757696","536540555136","1178991754368","1694431369632","1666320170800","1138590063040","536886720512","169503390200","33598294300","3669022072","161755320","797174","1"],["197553645440","1781336869248","7260029518912","17648849880960","28428801569312","31861676632928","25365389068592","14366796702400","5696744508456","1522370496360","254797766796","23075937208","792833094","2391498","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","79","601","5339","53861","607527","7560625","102637235","1506225085","23726435583","398852249097","7120170905995","134408217821205"],"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":[["0","-1"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165346,"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-a8162231721a7380"]},{"id":165347,"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-a8162231721a7380"]},{"id":165348,"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-a8162231721a7380"]},{"id":165349,"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-a8162231721a7380"]}],"region_exact":{},"incoming_edges":[{"id":"e-a8162231721a7380","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-0c5ba5f813be2bea","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":"47521f03b0dde58ceb673379febca24311acc534df4aad85a49203ff740b9f9f","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-28595a60eb57b37f","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of 2nd kind.","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-03f854ab79acf219","kind":"fingerprint_related","node_a_id":"n-0c5ba5f813be2bea","node_b_id":"n-8c58f795ab804834","fingerprint":"f6-078627b5c58f7a3e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-078627b5c58f7a3e"},"other_id":"n-8c58f795ab804834","other":{"id":"n-8c58f795ab804834","name":"mobius_x_over_x_plus_one of Triangle of B-analogs of Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-078627b5c58f7a3e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","16","38","24"]],"row_sums_preview":["1","3","13","79","601","5339","53861","607527"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d55b2cfe05a04774","steps":["x \u21a6 x/(x+1)"],"edges":["e-a5b6b9e544860989"],"text":"A039755 \u2192 x \u21a6 x/(x+1)"},"headline":"A039755 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-c5b90c39c5bc94d9","kind":"oeis_match","node_a_id":"n-0c5ba5f813be2bea","node_b_id":null,"fingerprint":null,"oeis_id":"A355165","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A355165","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A355165","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-0c5ba5f813be2bea","oeis_id":"A355165","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":34499}],"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,        1;\n  2:       6,        6,        1;\n  3:      24,       38,       16,        1;\n  4:     116,      256,      184,       44,        1;\n  5:     648,     1844,     1896,      824,      126,        1;\n  6:    4088,    14296,    19236,    12264,     3606,      370,       1;\n  7:   28640,   119160,   199088,   168804,    74936,    15798,    1100,      1;\n  8:  219920,  1063168,  2128960,  2265696,  1361256,   448160,   70176,   3288,    1;\n  9: 1832224, 10107152, 23642944, 30380608, 23163568, 10512872, 2670032, 317984, 9850, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+(2*n-1)*x)*P(n-1,x) - (2+4*x+2*x^2)*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*n - 1)*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*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*n - 1)*x)*v[i - 1] - (2 + 4*x + 2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-28595a60eb57b37f","steps":["x \u21a6 x+1"],"edges":["e-a8162231721a7380"],"text":"A039756 \u2192 x \u21a6 x+1"},"headline":"A039756 \u2192 x \u21a6 x+1","chain":null}