{"id":"n-0d23d1c7119dd8ef","revision":1,"first_index":0,"fingerprint":"f6-d86661683c23bbdc","name":"basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind.","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":[["12","21","9"],["60","141","108","27"],["360","1026","1071","486","81"],["2520","8262","10575","6615","2025","243"],["20160","73656","109386","84645","36045","8019","729"],["181440","723384","1205442","1089963","578340","180306","30618","2187"],["1814400","7778160","14224572","14515956","9053289","3538080","847098","113724","6561"],["19958400","91002960","179804772","202349232","143134047","66078747","19932318","3792258","413343","19683"],["239500800","1151910720","2430666144","2967605100","2324656260","1222347105","437424057","105304050","16336890","1476225","59049"],["3113510400","15693341760","35054392032","45870864732","39123346680","22864481145","9353554056","2681224821","528291720","68201595","5196312","177147"],["43589145600","229047315840","537841513728","747355282344","685339447716","437472776070","199543200219","65597809662","15439758543","2539697490","277353153","18068994","531441"],["653837184000","3566477174400","8754764653440","12823853776344","12522157562772","8618109984198","4305566331495","1582596745587","428389807131","84414737979","11779389765","1103094369","62178597","1594323"],["10461394944000","59025146342400","150775665978240","231445954381824","238826082333384","175456232435484","94743391256514","38238246923877","11602027150857","2635805229057","441714450177","52987679199","4304140659","212044959","4782969"],["177843714048000","1034811672652800","2740261760657280","4386908222425728","4754381262813000","3699234198403380","2137006348667190","934280371475451","311949202336200","79614770346540","15416561340180","2225933896914","232133428800","16517186280","717445350","14348907"],["3201186852864000","19160141249894400","52429146709789440","87185133285634944","98739587397911184","80849359359699840","49563816871219560","23228065732559688","8417926756477953","2368913473246320","516342415162860","86316494164992","10856203409142","993709639440","62465575140","2410616376","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["42","336","3024","30240","332640","3991680","51891840","726485760","10897286400","174356582400","2964061900800","53353114214400","1013709170073600","20274183401472000","425757851430912000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["12","21","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","3"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167499,"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-36015c56c1237184"]},{"id":167500,"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-36015c56c1237184"]},{"id":167501,"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-36015c56c1237184"]}],"region_exact":{},"incoming_edges":[{"id":"e-36015c56c1237184","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-0d23d1c7119dd8ef","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":"2cab8e8391f1df5fe878b4d775bd0fd1c5e433e8cbedab7af9a961f71c2bf173","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-c237629cf5fdbb07","role":"sequence","name":"Unsigned 2-Stirling numbers of the first 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-7589c2be95265299","kind":"fingerprint_related","node_a_id":"n-f617181cb238a8cd","node_b_id":"n-0d23d1c7119dd8ef","fingerprint":"f6-d86661683c23bbdc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d86661683c23bbdc"},"other_id":"n-f617181cb238a8cd","other":{"id":"n-f617181cb238a8cd","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","first_index":0,"fingerprint":"f6-d86661683c23bbdc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["12","21","9"]],"card_rows":[["1"],["3","3"],["12","21","9"],["60","141","108","27"]],"row_sums_preview":["1","6","42","336","3024","30240","332640","3991680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-914700b1060feeb2","steps":["basis A013610"],"edges":["e-bb12b9c216f60d78"],"text":"A136124 \u2192 basis A013610"},"headline":"A136124 \u2192 basis A013610","chain":null}},{"id":"l-dc0376d0c0d59dff","kind":"oeis_match","node_a_id":"n-0d23d1c7119dd8ef","node_b_id":null,"fingerprint":null,"oeis_id":"A001725","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":5,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001725","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001725","relation":"match_row_sums"}],"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:         12,          21,           9;\n  1:         60,         141,         108,          27;\n  2:        360,        1026,        1071,         486,          81;\n  3:       2520,        8262,       10575,        6615,        2025,         243;\n  4:      20160,       73656,      109386,       84645,       36045,        8019,        729;\n  5:     181440,      723384,     1205442,     1089963,      578340,      180306,      30618,       2187;\n  6:    1814400,     7778160,    14224572,    14515956,     9053289,     3538080,     847098,     113724,      6561;\n  7:   19958400,    91002960,   179804772,   202349232,   143134047,    66078747,   19932318,    3792258,    413343,    19683;\n  8:  239500800,  1151910720,  2430666144,  2967605100,  2324656260,  1222347105,  437424057,  105304050,  16336890,  1476225,   59049;\n  9: 3113510400, 15693341760, 35054392032, 45870864732, 39123346680, 22864481145, 9353554056, 2681224821, 528291720, 68201595, 5196312, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+4+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 12 + 21*x + 9*x^2.","mathematica":"P[0, x_] := 12 + 21*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(n + 4 + 3*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] = 12 + 21*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 4 + 3*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-c237629cf5fdbb07","steps":["basis A013610"],"edges":["e-36015c56c1237184"],"text":"A143491 \u2192 basis A013610"},"headline":"A143491 \u2192 basis A013610","chain":null}