{"id":"n-464957f72336fbd0","revision":1,"first_index":1,"fingerprint":"f6-24e672df326049ba","name":"obreschkoff_combination of Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","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":[["3","3"],["6","24","18"],["12","138","288","162"],["24","726","3132","4374","1944"],["48","3702","29412","76302","79704","29160"],["96","18654","257508","1104030","2031480","1691280","524880"],["192","93558","2171988","14461902","41689080","59223960","40940640","11022480"],["384","468366","17937252","178628814","757189944","1632143520","1884844080","1113270480","264539520"],["768","2342982","146308212","2126352222","12744204984","39104289000","67075464960","65186946720","33596519040","7142567040"],["1536","11717214","1184523588","24706648350","203935096440","855935987760","2045386501200","2907884538720","2437996216320","1114240458240","214277011200"],["3072","58590678","9546491988","282433844142","3151571130360","17609938238520","56314577803680","109834460915760","133177130553600","98138871129600","40284078105600","7071141369600"],["6144","292962606","76723479972","3192690713454","47511201954744","346642517010240","1443270444366960","3708798734939760","6098632456371840","6441809787705600","4233256633267200","1576864525420800","254561089305600"],["12288","1464831462","615545615412","35810109167742","703469115927864","6605590818495240","35104974193523520","115611050235320640","247575613504222080","351475560165502080","328718505895718400","194887727287545600","66440444308761600","9927882482918400"],["24576","7324194174","4933153912068","399451111383870","10278288933003000","122847080653337040","821000118603384720","3396258613476368640","9211631156757469440","16877332809413556480","21063258993628051200","17668781149622803200","9540695066084582400","2998220509841356800","416971064282572800"],["49152","36621044598","39509176461588","4438360610431182","148689458398648440","2242574705101774680","18631249823827761120","95354950600627557840","321012616799127052800","738156692705444812800","1180344272078504332800","1313494887026523801600","998622533897635046400","495014148480794342400","144271988241770188800","18763697892715776000"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["6","48","600","10200","218328","5627928","169603800","5849022360","227124995928","9805558700568","465873808176600","24154617097504920","1357087245127562328","82127679242886000408","5326062779616879082200"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162848,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-2dc56bf9965267b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-2dc56bf9965267b7","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-464957f72336fbd0","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"8b6ae18946e55480f13808bc948f87270d78b2e7c63e250c6ddcadd61abb8e01","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-170093e6e241015d","role":"sequence","name":"Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-170093e6e241015d","role":"sequence","name":"Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    3,        3;\n   2:    6,       24,         18;\n   3:   12,      138,        288,         162;\n   4:   24,      726,       3132,        4374,         1944;\n   5:   48,     3702,      29412,       76302,        79704,        29160;\n   6:   96,    18654,     257508,     1104030,      2031480,      1691280,        524880;\n   7:  192,    93558,    2171988,    14461902,     41689080,     59223960,      40940640,      11022480;\n   8:  384,   468366,   17937252,   178628814,    757189944,   1632143520,    1884844080,    1113270480,     264539520;\n   9:  768,  2342982,  146308212,  2126352222,  12744204984,  39104289000,   67075464960,   65186946720,   33596519040,    7142567040;\n  10: 1536, 11717214, 1184523588, 24706648350, 203935096440, 855935987760, 2045386501200, 2907884538720, 2437996216320, 1114240458240, 214277011200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) + (3*x+3*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 3 + 3*x.","mathematica":"P[1, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*x)*P[n - 1, x] + (3*x + 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 3 + 3*x; for(n = 2, N, my(i = n); v[i] = (2 + 3*x)*v[i - 1] + (3*x + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-170093e6e241015d","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-2dc56bf9965267b7"],"text":"A225472 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A225472 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}